Genetic redundancies enhance information transfer in noisy regulatory circuits

Guillermo Rodrigo1 and Juan F. Poyatos${}^{2}\dagger$

1 Instituto de Biología Molecular y Celular de Plantas, CSIC – UPV, 46022
Valencia, Spain, 2 Logic of Genomic Systems Laboratory (CNB-CSIC), 28049
Madrid, Spain.

† Corresponding author. Logic of Genomic Systems Laboratory (CNB-CSIC), Darwin
3, Campus de Cantoblanco, 28049 Madrid, Spain.

E-mail<EMAIL_ADDRESS>
## Abstract

Cellular decision making is based on regulatory circuits that associate signal
thresholds to specific physiological actions. This transmission of information
is subjected to molecular noise what can decrease its fidelity. Here, we show
instead how such intrinsic noise enhances information transfer in the presence
of multiple circuit copies. The result is due to the contribution of noise to
the generation of autonomous responses by each copy, which are altogether
associated with a common decision. Moreover, factors that correlate the
responses of the redundant units (extrinsic noise or regulatory cross-talk)
contribute to reduce fidelity, while those that further uncouple them
(heterogeneity within the copies) can lead to stronger information gain.
Overall, our study emphasizes how the interplay of signal thresholding,
redundancy, and noise influences the accuracy of cellular decision making.
Understanding this interplay provides a basis to explain collective cell
signaling mechanisms, and to engineer robust decisions with noisy genetic
circuits.

## INTRODUCTION

The biochemistry of cells determines the operation of biological circuits.
This biochemistry is inevitable noisy [McAdams and Arkin, 1997, Elowitz et
al., 2002, Lestas et al., 2010] what immediately suggests a limitation to the
reliable function of these circuits, and thus many early studies examined how
the problem of achieving correct operation could nevertheless be solved.
Mechanisms such as kinetic proofreading [Hopfield, 1974], or integral feedback
control [Yi et al., 2000] emerged then as some fundamental solutions. One
might ask, on the other hand, to what extent noise could indirectly represent
an advantage. An example is found when cell populations, in which noise leads
to phenotypic variability, display heterogeneity in stress responses that
represent a crucial element for survival, e.g., [Bishop et al., 2007].

In a more direct situation, noise can turn into a indispensable ingredient to
facilitate new classes of behaviors not achievable otherwise [Balaban et al.,
2004, Süel et al., 2006, Acar et al., 2008, Turcotte et al., 2008, Raj et al.,
2010]. These valuable behaviors are typically related to cellular decisions,
which essentially involve changes in the expression phenotype. Specific
biological circuits were consequently shown to employ noise to induce the
expression of transient phenotypes [Süel et al., 2006], or to switch among
distinct stable states [Acar et al., 2008]. That many of these probabilistic
dynamics relate to systems whose actions are susceptible to limiting signal
values [Feinerman et al., 2008] emphasizes the connection between noise,
cellular decisions, and threshold response circuits.

The beneficial aspect of noise also forces us to revisit some of the early
arguments on the relationship between stochasticity and the structure of
biological systems [Lerner, 1954, McAdams and Arkin, 1999]. In particular, the
existence of genetic redundancies was typically interpreted as a mean to
enhance reliability of operation (i.e., noise as a disruptive element). This
role appeared in consequence as a plausible rationale for the evolutionary
maintenance of several copies of a gene or circuit [Nowak et al., 1997].
Instead, we focus here on redundancy as a genetic architecture that, when
coupled to the effect of noise in threshold response circuits, enables unique
information-processing functions.

We examined this issue within the precise framework of information theory.
Biological circuits are in this way interpreted as communication channels, in
which an input signal ($x$) originates –as a result of a cellular decision– an
expression output ($y$), with a given probability (Fig. 1A). The uncertainty
on the input signal is then reduced by the decision process, whose set of
outcomes tells us about the input distribution [Levchenko and Nemenman, 2014,
Bowsher and Swain, 2014]. This association is properly quantified by the
mutual information (MI), an information-theoretic measure describing the
correlation between the input signal and the output phenotype (Fig. 1A).
Notably, this framework was recently exploited to quantify the functionality
of transcriptional regulatory elements [Tkacik et al., 2008, Libby et al.,
2007, Yu et al., 2008], the accuracy of cell location during developmental
processes [Dubuis et al., 2013], and the maximal information transmission
capacity of noisy signaling pathways [Cheong et al., 2011, Hansen and O’Shea,
2015]. The relevance of redundancies was already manifested in some of these
results.

Here, we first illustrate how intrinsic noise (from stochastic biochemical
reactions) can help to gain information. We then show that information
transfer can be amplified, if the combined response of multiple genetic units
is considered. The reported amplification is shown to rely on the presence of
different factors that contribute to generate variability in the individual
response of each unit, like intrinsic noise or genetic heterogeneity (i.e.,
differences in the biochemical properties). This variability helps to enlarge
the capacity of the global output to represent the input distribution. In
contrast, we also discuss how factors reducing variability in the responses,
like a noise source common to all units (extrinsic noise) or regulatory cross-
talk, eventually mitigate the gain.

## RESULTS

### Intrinsic noise can amplify information transfer

We first analyzed a minimal regulatory circuit implemented by a gene (whose
expression we denote as $y$) autoactivating transcriptionally its own
production [Wall et al., 2004]. This is a genetic implementation of a
threshold device that, by acting deterministically, becomes activated only if
the input signal $x$ crosses a particular limit (Fig. 1B). When the signal is
stochastic, the response depends of course on the relationship between this
threshold and the mean (and variance) of the underlying distribution $P(x)$
(considered for simplicity as a uniform distribution; Fig. 1B). A symmetric
distribution centered on the threshold would thus originate equally likely the
two output values (OFF/ON) (i.e., one bit of information); while the same
distribution centered above/below the threshold would produce biased responses
(i.e., less than one bit of information).

However, the previous behavior can be affected by the extensive noise sources
acting on biological circuits [McAdams and Arkin, 1997, Elowitz et al., 2002,
Lestas et al., 2010]. One could ask then to what extent the circuit is
reliably representing the signal. To quantify how much information the
response conveys about the input, we made use of MI [Levchenko and Nemenman,
2014, Bowsher and Swain, 2014] (Fig. 1A). We thus computed the response to a
number of signals drawn from a fixed distribution and strength of intrinsic
noise (black dots in subpanels of Fig. 1C), which allowed us to quantify the
value of MI. This value changes with noise [main plot in Fig. 1C; the mean of
$P(x)$ is above the threshold, red distribution in Fig. 1B]. For weak noise
levels, the circuit works essentially as a deterministic switch, it is always
$y=$ ON as $x>$ threshold. For strong noise levels, the device cannot
distinguish signal fluctuations, then its behavior is essentially random. In
both cases, the information that the gene processes is limited (subpanels of
Fig. 1C, red curves denote the averaged stimulus-response profiles). But MI
presents a maximum for an intermediate noise level. In this regime, the
circuit can express its two possible states due to noise (i.e., low values of
$x$ can cross the threshold) [Gammaitoni, 1995], what precisely contributes to
a better representation of the input signal (see also Fig. S1); a
characteristic behavior of noisy nonlinear systems known as stochastic
resonance (SR) [Gammaitoni et al., 1998].

Moreover, SR disappears when the mean of $P(x)$ is close to the threshold, as
stochasticity is now not required to reach the two possible states. In this
case, noise always reduces information transfer (Fig. 1D, curves for $N=1$).
Note here how MI does exhibit an upper limit of 1 bit when the mean of $P(x)$
exactly matches the threshold, and the circuit is noiseless. MI decreases with
noise because signal values above/below the threshold originate in some cases
stochastic crossings (e.g., $y=$ ON when $x<$ threshold), and the information
content in absence of noise is already high (note in contrast that, in the
scenario of SR, MI was very low in absence of noise). Additionally, Figure 1D
displays a situation in which a maximum in MI is nevertheless observed (curves
for $N=2$). This is obtained by increasing the number of devices processing
the same input, with $y$ representing in this case the sum of all individual
outputs; a phenomenon called suprathreshold SR [Stocks, 2000]. What is
apparent here is that redundancy boosts information transfer, given a fixed
noise level.

### Genetic redundancy enhances information transfer

The addition of extra copies of the threshold device, i.e., genetic
redundancy, appears then as a potential mechanism to increase the transmission
of information in the presence of intrinsic noise. Consider, for instance, a
situation in which two devices read in parallel the same input signal,
assuming again two possible values of gene expression for each unit. The
overall output alphabet [Shannon, 1948] consists of three letters: {0 (both
copies OFF), 1 (one OFF the other ON), 2 (both ON)}. The new alphabet is
linked, of course, to the action of independent (intrinsic) noise sources
acting on the two genes, which allows each device to produce an autonomous
response (with noise-induced threshold crossings). The sum of individual
responses would give, accordingly, a global output distribution $P(y)$
constituted by three peaks. The extended alphabet helps therefore to enlarge
the capacity of the output to represent the input variability; in other words,
it contributes to linearize the averaged stimulus-response profile (Fig. 2A,
see also Fig. S2).

Both the number of units and the type of nonlinearity influence the increment
of information transfer. In Figure 2B, we introduced three different threshold
devices [Wall et al., 2004] to show how MI increases with redundancy. For each
type, MI relative to the case of no redundancy (i.e., a single unit) was
plotted. Specifically, we examined a simple regulated unit, a bistable
expression system implemented through a positive feedback (the architecture
that we discussed before), and an excitable device constituted by interlinked
positive and negative feedbacks [implemented as the one linked to transient
differentiation in Bacillus subtilis [Süel et al., 2006]]. The output of all
these devices is given by a continuous variable representing gene expression
(note that the response was previously regarded as OFF/ON). This allowed
identifying discrepancies in terms of MI among different gene regulatory
circuits. In particular, the largest amplification of information content
corresponds to those devices whose actions ultimately rely on discontinuous
transitions (i.e., the bistable and excitable systems). Out of these two
systems, the excitable one presents comparatively larger amplification,
although only observed for relatively large arrays. This is associated to the
fact that, in this system, the response is entirely binary even in presence of
noise: either the signal triggers a response or not (Fig. S3). Moreover, the
gain in information transfer is much lower for the simple regulated system. In
this case, the stimulus-response profile is continuous (i.e., no discontinuous
transition is produced) what entails that one unit already has the capacity to
reach a relatively large output alphabet. The contribution of redundancy is
therefore always much higher in analog-to-digital than in analog-to-analog
signaling circuits (and provided they are noisy).

### Input signal distribution shapes information transfer

The specific distribution of the signal impinging on the genetic circuits can
encode specific environmental or genetic conditions [Sharpe et al., 2001],
which can further modulate the enhancement of information transfer. We first
analyzed the effect of the shape of $P(x)$. We considered three different
signals acting on the array of threshold devices. A normal distribution
contributes in higher extent to increase MI with genetic redundancy (Fig. 3A).
For this distribution, the mass of $x$ values is closer to the threshold,
existing more chances to subvert the deterministic decision of the device due
to noise. We then analyzed the effect of the relationship between the
threshold and the signal mean. When the mean of $P(x)$ is equal to the
threshold, a higher increase of MI with genetic redundancy is observed (Fig.
3B). Arguably, if the mass of $x$ values is equally distributed above/below
the threshold, there exists again more chances for noise-induced threshold
crossings. Fine-tuning of the parameters characterizing $P(x)$ contributes
thus to a better representation of the input signal by the global output
response.

### Extrinsic noise and cross-talk limit information transfer

The most important constraint for the gain in information associated to the
previous redundant systems is the independence between the noise sources. When
these are correlated, $P(y)$ becomes more sharply peaked around a small subset
of possible responses (i.e., the output alphabet is more limited; Fig. 4A).
This applies to biological circuits that, in addition to intrinsic noise, also
integrate the effect of extrinsic fluctuations [Elowitz et al., 2002, Swain et
al., 2002]. This type of noise affects all genetic devices in the same manner
what eventually correlates individual outputs. Figure 4B shows how (relative)
MI decreases with the strength of extrinsic noise in an array of five bistable
units. Note however that this redundant architecture still exhibits, for
different extrinsic noise levels, a larger MI with respect to the nonredundant
case (inset of Fig. 4B).

Despite the independence of the noise sources, the presence of cross-talks
between devices can similarly lead to correlations in the individual gene
responses. In a genetic context, one could imagine two independent
transcription factors sharing recognition domains [Masquilier and Sassone-
Corsi, 1992]. One could also imagine a second unit recently emerged by
duplication, and that no process of neofunctionalization yet occurred
[Hittinger and Carroll, 2007]. Figure 4C indeed shows a decay in (relative) MI
for a system of two units when cross-talk between them increases (simulations
done without accounting for extrinsic noise). In this case, the activation of
one unit drags the activation of the other, biasing again the output alphabet
(inset of Fig. 4C). Of note, the decay profile in MI is qualitatively
different in the two scenarios. Addition of extrinsic noise contributes to
limit information transfer in a progressive manner since it increasingly
coordinates responses. In the second scenario, outputs are correlated once a
relatively specific cross-talk range is reached what is reflected in a more
abrubt decay.

### Heterogeneity also contributes to enhance information transfer

A complementary source of individuality in information processing could be
linked to the heterogeneity within the collection of threshold devices. In the
context of genetic circuits, this corresponds to the variability in promoter
strengths, ribosome-binding sites, proteins half-lives, or protein-DNA binding
affinities; all factors that in effect modify threshold values or output
responses. Adjusting for each device the values of the biochemical parameters
of the model can capture this variation [Mayo et al., 2006]. We specifically
explored the implication of threshold heterogeneity in the array of five
bistable units.

Notably, we observed again a resonance in information transfer, but this time
as a function of the degree of heterogeneity (Fig. 5). While moderate levels
of heterogeneity allows regulatory circuits to encode complementary aspects of
the input signal, hence enhancing information transfer, larger levels of
variation originates noise-induced threshold crossings over the whole input
range, which is detrimental to represent $P(x)$ with $P(y)$ (note that these
crossings occur in a narrower range when less variation is considered).
Moreover, since both intrinsic noise and heterogeneity contribute to increase
the transmission of information, we also explored to what extent these two
sources of individuality work independently [Hunsberger et al., 2014]. We
found that intrinsic noise mitigates the increase in MI due to threshold
variability (inset of Fig. 5). Intuitively, higher noise levels make
indistinguishable those regulatory variations in terms of gene expression.

## DISCUSSION

Binary decisions implemented by means of threshold devices appear in many
engineering and physical systems, and have been extensively studied in
relation to the detection and transmission of signals. While noise was
commonly considered harmful in many of these scenarios, some work
alternatively identified circumstances in which its presence enhances
performance [Gammaitoni et al., 1998, McDonnell and Ward, 2011]. In Biology,
both the stochastic nature of biochemical reactions and the typical occurrence
of thresholds –linked, for instance, to cell fate determination– also
anticipates the possibility of beneficial effects. This specifically applies
to the case of gene regulatory circuits, in which molecular stochasticity acts
in many cases as a core determinant of function [Eldar and Elowitz, 2010].

In this work we discussed in detail the benefits of intrinsic molecular noise
when multiple threshold regulatory circuits process a common signal. This
system exhibits a resonance phenomenon known as suprathreshold SR [Stocks,
2000]. The effect establishes the benefit of the noise-induced uncoupling of
the action of each unit. This advantage is manifested as well in a more linear
relation between stimulus and response, a type of dose-response alignment that
could be important in how precise extracellular conditions determine cell
responses, and that was previously associated to negative feedbacks [Yu et
al., 2008]. Our functional analysis therefore reveals redundancies not only as
a genetic architecture contributing to robustness [Kafri et al., 2006, Keane
et al., 2014], or to the adaptation to novel environments through the increase
of gene expression levels [Riehle et al., 2001, Gresham et al., 2008], but
also as a mechanism increasing the capacity to transmit reliable information
(Fig. 6). We suggest that this aspect could selectively contribute to the
evolutionary maintenance of genetic redundancy. That multiple signaling
pathways in Saccharomyces cerevisiae overlap supports this hypothesis [van
Wageningen et al., 2010].

The balance of intrinsic/extrinsic noise also plays an important part to
condition the amount of information transferred (Fig. 6). Cells implementing
regulatory circuits with few representative molecules or living in rich
environments would shift this balance towards intrinsic noise [Volfson et al.,
2006]. Beyond this genetic/environmental tuning, cellular systems could avoid
the loss of information, due to extrinsic noise, when the signal operates
dynamically rather than statically [Selimkhanov et al., 2014]. Note that here
we considered a static operation. Our results further emphasize how
heterogeneity and cross-talk among redundant copies play opposite roles in the
maintenance of information content (Fig. 6). One could thus interpret the
action of several parallel signaling pathways, each conveying approximately 1
bit of information, as heterogeneous copies of an effective threshold device
what enhances information transmission, e.g., this was observed in pathways
for the growth factor-mediated gene expression [Uda et al., 2013].

That a global response –the sum of individual responses, in this case–
implemented by parallel processing units could lead to better performance than
that of the individual components was proposed in early models of computing,
and can indeed be observed at different levels of biological organization:
from genes (this work), to living cells [Cheong et al., 2011], to social
organisms [Conradt and Roper, 2005]. In addition, ideas on redundancy and
heterogeneity when mounting unreliable components were already present in the
initial development of fault-tolerant computation and communication [von
Neumann, 1956, Moore and Shannon, 1956], and also permeate to many biological
scenarios. Our work substantiates the implications of these notions in
cellular decision making by natural [van Wageningen et al., 2010] and
synthetic [Dueber et al., 2007] molecular circuits, and contributes to
exemplify how the application of concepts from information theory could lead
to a more precise and quantitative understanding of cellular systems.

## THEORETICAL PROCEDURES

### Modeling noisy regulatory systems

We considered a redundant system consisting of $N$ different transcriptional
units, each of them activated by the input signal ($x$). The model for the
$i$-th unit reads

$\frac{dy_{i}}{dt}=f(y_{i},x)+q_{i}(y_{i},x)\xi_{i}(t),$ (1)

where expression ($y_{i}$) and time are appropriately rescaled to have a
dimensionless model. To model different regulatory systems (simple, bistable
or excitable), we modified the function $f$ (see the Supplement for details of
functional forms and parameter values). $\xi_{i}$ is a stochastic process that
has mean $0$ and is $\delta$-correlated. Noise amplitude is given by the
square root of the sum of propensities.
To account for extrinsic noise, we introduced a new stochastic process
($\xi_{ex}$), common to all units, in Eq. (1) as

$\frac{dy_{i}}{dt}=f(y_{i},x)+q_{i}(y_{i},x)\xi_{i}(t)+q_{ex}\xi_{ex}(t).$ (2)

The correlation time of extrinsic noise is of the order of the cell cycle (the
mean is also $0$). For simplicity, we here supposed a system implemented with
short-lived proteins, so we can assume that $\xi_{ex}$ is constant within the
time window that the system needs to reach its steady state upon receiving the
perturbation $x$.
To account for certain cross-talk between the different units of the system,
we followed a perturbative approach to obtain

$\frac{dy_{i}}{dt}=f(y_{i},x)+q_{i}(y_{i},x)\xi_{i}(t)+\varepsilon\sum_{j=1,j\neq
i}^{N}y_{j},$ (3)

where $\varepsilon$ quantifies the degree of cross-talk. For simplicity, we
assumed $q_{i}$ not to be dependent on $y_{j}$ for $j\neq i$.
To account for heterogeneity, we included variations in the threshold values
of the different units of the system. This was modeled by introducing a
Gaussian random number $\omega$ of mean $1$, being its standard deviation the
degree of heterogeneity [$f(y_{i},x,\omega_{i})$ for the $i$-th unit, where
$\omega_{i}$ is a realization; see details in the Supplement]. When accounting
for cross-talk or heterogeneity, only the intrinsic noise was considered.

### Input and output variables

Here, we contemplated that the regulatory system is initially in a steady
state in which there is no input signal (i.e., $x=0$ for $t<0$). We then
considered that $x$ becomes activated (at $t=0$); as a step function in the
case of the simple and bistable systems, or as a pulse function (for one unit
of normalized time) in the case of the excitable system. The value of $x$ for
$t>0$ was modeled as a random number following a given probability
distribution. This models an input signal whose value can fluctuate according
to upstream processes, environmental changes or molecular noise. We also
regarded that the array of genes is able to perceive this signal to change the
individual expression levels ($y_{i}$) accordingly. The output was calculated
at steady state. We assumed that the signal fluctuations occur at a frequency
that allows the genetic circuit to respond against the current signal value.
The change in gene expression due to signaling was defined by $\Delta
y_{i}=y_{i}(x)-y_{i}(x=0)$. The total differential gene expression of the
redundant system can be written as $\Delta y=\sum_{i=1}^{N}\Delta y_{i}$. In
case of the excitable system, because the response is transient, we considered
a Boolean function operating on $y_{i}$, setting $1$ if the unit was excited
or $0$ if not. In the first section of the paper (Fig. 1), the gene expression
level ($y_{i}$) was treated as a Boolean variable (OFF/ON). In the subsequent
sections (Figs. 2 – 5), it was treated as a continuous variable.
In addition, we studied different distributions of $x$. We mainly included a
uniform distribution covering two orders of magnitude. In Figs. 1 and 3B, we
analyzed the effect of the mean of the distribution, and the values were
$0.001$ (equal to the threshold value), $0.005$ and $0.01$. In Fig. 2, the
mean was fixed to the threshold value, i.e., $0.001$ in the case of the
bistable system, $1$ in the simple regulated unit, and $0.9$ in the excitable
system. In Figs. 4 and 5, concerning to the bistable system, the mean of the
distribution of $x$ was $0.005$. We additionally considered different forms of
the distribution. In Fig. 3A, we analyzed the effect a normal or beta
distributions in log scale, with the mean equal to the threshold value.

### Quantification of information transfer

We used mutual information (${\cal I}$) as a quantitative metric to describe
how the global output response of a single cell is sensitive to different
concentrations of the input signal. This extends the quantification by the
averaged stimulus-response profile. To calculate ${\cal I}$, we performed
$10^{4}$ realizations of the pair ($x$, $y$) and then we solved numerically
the following integral

$\displaystyle{\cal I}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{+\infty}P_{\Delta y}(s)\log_{2}P_{\Delta
y}(s)ds+\int_{-\infty}^{+\infty}P_{\log x}(r)$ (4) $\displaystyle\times$
$\displaystyle\int_{-\infty}^{+\infty}P_{\Delta y|\log x}(s)\log_{2}P_{\Delta
y|\log x}(s)dsdr,$

where we considered $\log x$ as input and $\Delta y$ as output variables. By
using the Fokker-Planck equation, we calculated the probability that a unit
has a given gene expression level (see more details in the Supplement).

## Acknowledgments

We thank M.A. Fares and A. Couce for a critical reading of the manuscript.
This work was supported by grants (to GR and JFP) from the Spanish Ministry of
Economy and Competitiveness.

## Author Contributions

G.R. and J.F.P. performed research and wrote the manuscript.

## Competing Financial Interests

The authors declare no competing financial interests.

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Figure 1: Intrinsic noise can increase or decrease information transfer in
threshold genetic systems. (A) A noisy channel is characterized by the mutual
information (MI) ${\cal I}$ of the output ($y$) given an input ($x$). MI
quantifies the dependence between input and output distributions, $P(x)$ and
$P(y)$ respectively. This could be estimated by a correlation coefficient, but
this measure cannot discriminate some associations better captured by MI. In
the cartoon we show two cases with the same correlation (whose value we
represented here by the eccentricity of the ellipses) but different MI. (B)
The channel can describe a gene autoactivating its own expression ($y$) in a
bistable OFF/ON manner what represents a simple example of threshold
regulatory circuit. Information transfer depends on the relationship between
$x$ and the threshold value of activation ($x$ and $y$ are presented in
arbitrary units). Three instances of $P(x)$ are shown (uniform distributions
with different means; the blue one corresponds to a mean equal to the
threshold value). When the signal is always beyond the threshold (red
distribution) the circuit exhibit a nonzero MI only when it works
stochastically (note the two different output distributions). Here we
considered a binary response (OFF if $y<1$, ON otherwise). (C) Resonance in MI
as a function of the strength of intrinsic noise (see Theoretical Procedures)
for the red $P(x)$ in (B). Each subplot displays the responses of the device
to $10^{4}$ signal values drawn from the described distribution (black dots),
and the corresponding averaged stimulus-response profile (red curve), for
three explicit noise levels. The maximum in MI occurs when the averaged
stimulus-response profile is more linear (Fig. S1). (D) Other signal
distributions, in which intrinsic noise always reduces MI, can nevertheless
exhibit a resonance when the combined response of several units is considered
(we show here the case of duplicated threshold devices; $N=2$). Colors
correspond to those distributions shown in (B).

Figure 2: Genetic redundancy amplifies information transfer in threshold
genetic systems. (A) Input/output distributions depicting information
transfer. The input distribution (in yellow) is assumed to be uniform. Output
distributions (in gray) illustrate the processing of the signal $x$, either
through a single copy of the threshold device (left) or an array of multiple
redundant copies (right). In the latter case, each unit of the array receives
the same signal and the output $y$ is the sum of all the individual responses.
Redundancy effectively enlarges the alphabet of the response. This is
reflected in the output distribution, and also in the linearization of the
averaged stimulus-response profile (black curve). (B) (Left) Array of $N$
threshold devices whose constituent units correspond to (1) a simple regulated
unit, (2) a bistable circuit implemented with a positive feedback, and (3) an
excitable circuit constituted by two interlinked positive and negative
feedback loops. (Right) Dependence of mutual information (MI) with the number
of units ($N$) for each of these systems. MI relative to the case $N=1$. A
uniform signal distribution with mean equal to the threshold value was
considered. In the case of noiseless units, MI does not increase with extra
copies (independently of the type of unit; dashed line). See the Supplement
for details of the model of each circuit.

Figure 3: The distribution of the signal modulates the increase of information
transfer due to genetic redundancy. (A) Effect of the form of the distribution
on MI: (1) uniform (covering two orders of magnitude), (2) lognormal (with
standard deviation equal to 2/3), and (3) beta in log scale (with the two
shape parameters equal to 1/3). In all cases, the mean of the distribution is
equal to the threshold value. (B) Effect of the mean of the distribution (here
uniform) on MI: (1) equal to the threshold value, (2) and (3) deviated from
the threshold value. We considered as threshold device a bistable unit
implemented with a positive feedback in all plots (see Theoretical
Procedures).

Figure 4: Extrinsic noise and cross-talk among redundant copies limit
information transfer. (A) Input/output distributions depicting information
transfer. In this case, correlation among individual gene responses due to
extrinsic noise or cross-talk reduces the response alphabet, and generates a
less linear averaged stimulus-response profile (black curve, see Fig. 2A for
comparison). (B) Dependence of mutual information (MI) with the strength of
extrinsic noise (see Theoretical Procedures). Relative MI is with respect to
absence of extrinsic noise. For this plot, we considered a system of $N=5$
bistable units implemented with positive feedback. The inset shows a direct
comparison between $N=1$ and $N=5$, emphasizing that MI increases with $N$.
(C) Dependence of MI with the degree of cross-talk for the same regulatory
system, but now constituted by $N=2$ units. Relative MI is with respect to the
situation without cross-talk. The inset presents the marginal probability
distribution of gene expression of one unit ($y_{1}$) in the absence and
presence of cross-talk (parameterized by $\varepsilon$=0 and
$\varepsilon$=0.01, respectively; see Theoretical Procedures) for the mean
value of the input signal ($x$). Note that when the units are coupled, gene
expression becomes unimodal (dashed curve).

Figure 5: Genetic heterogeneity among redundant copies leading to functional
variability improves information transfer. Variation in the biochemical
features of the constituent threshold devices (here bistable units, $N=5$; see
Theoretical Procedures) leads to a maximum in mutual information (MI). The
inset indicates the peak differential MI (i.e., the difference between the
largest value of MI with heterogeneity and the value of MI without it) for
varying noise levels. This reveals how a situation of stronger intrinsic noise
contributes to reduce the improving effect on MI of heterogeneous units (the
main plot corresponds to an intrinsic noise amplitude equal to 0.16).

Figure 6: Model of information transfer in gene regulatory circuits. Intrinsic
noise, genetic redundancy, and heterogeneity increase the transmission of
information (by strengthening the capacity of the global output to represent
the input variability), whilst extrinsic noise and cross-talk among redundant
units become limiting factors (by correlating the individual outputs of the
units).


INTERNATIONAL JOURNAL OF GEOMETRY

Vol. 3 (2013), No. 1, 5 - 14

FROM TWO SIMPLE PROBLEMS TO

THE CONNECTION OF SPECIAL POINTS

NGUYEN TRAN HUU THINH

Abstract. Both the USA TST 2008 [1] and the ELMO Shortlist 2013 [2] suggested
two issues that are connected to fixed points. These problems provide a strong
linkage between the various attributes of specific points in a triangle. In
this article, we will first investigate various theorems concerning the fixed
points that have been presented, and then we will demonstrate how those points
are connected to a few triangle centers.††Keywords and phrases: Fixed point,
Triangle, Brocard point, Hagge circle ††(2020)Mathematics Subject
Classification: 51M04, 51-08 ††Received: 1.01.2013. In revised form:
13.06.2013. Accepted: 10.09.2013.

##  1\. Introduction

The fixed-point problems have been seen in a great number of international
mathematical olympiad competitions [4]. The generalization of these problems
in this area, as well as the features of these problems, leave behind a
plethora of notable results [10]. In this work, we talk about some of those
problems and show the relationship to a few of different kinds of unique
triangle centers.

In USA Team Selection Test 2008 [1], there is a fixed-point problem as
following:

######  Problem 1.

Given triangle $ABC$, centroid $G$. Let $P$ be a variable point on segment
$BC$. Let $Q,R$ be points on $AC,AB$ such that $PQ\parallel AB,PR\parallel
AC$. Prove that $(AQR)$ passes through a fixed point $X$ such that
$(AB,AG)=(AX,AC)$.

Figure 1. Proof of Problem 1

Proof. Let $\omega_{1}$ be the circle which passes $A,B$ and tangents to $AC$.
Let $\omega_{2}$ be the circle which passes $A,C$ and tangents to $AB$. Let
$X$ be intersection of $\omega_{1}$ and $\omega_{2}$, so $X$ is fixed. Since
two triangle $ABX$ and $AXC$ are homothetic, hence
$\frac{AX}{BX}=\frac{AC}{AB}$. On the other hand,
$\frac{AC}{AB}=\frac{PR}{BR}=\frac{AQ}{BR}$ so $\frac{AX}{BX}=\frac{AQ}{BR}$,
also $(BR,BX)=(AX,AX)$ hence two triangle $BRX$ and $AQX$ are homothetic,
therefore $(RB,RX)=(QA,QX)$, following that $A,Q,X,R$ are cyclic. Moreover, we
have

$\frac{\sin(AB,AX)}{\sin(CA,CX)}=\frac{\sin(AB,AX)}{\sin(BA,BX)}=\frac{BX}{AX}=\frac{AB}{AC}$

so $AX$ is the symmedian line of triangle $ABC$, which means
$(AB,AG)=(AX,AC)$. The problem has been solved.

With this configuration, ELMO Shortlist 2013 [2] provide the following problem
using the antiparallel lines instead of parallel ones:

######  Problem 2.

Given triangle $ABC$ and a centroid $G$. Let $P$ be a variable point on
segment $BC$. Let $Q,R$ respectively be points on $AC,AB$ such that $PQ,PR$
are antiparallel lines of $AB,AC$. Prove that $(AQR)$ passes through a fixed
point.

Figure 2. Proof of Problem 2

Proof. Let $\gamma_{1}$ be the circles passes $A,B$ and tangents to $BC$. Let
$\gamma_{2}$ be the circle passes $A,C$ and tangents to $BC$. Let $Y$ be
intersection of $\gamma_{1}$ and $\gamma_{2}$, so $Y$ is fixed. Let $K$ be
intersection of $BQ$ and $CR$. We hav

$(RK,RA)=(PC,PA)=(PB,PA)=(QK,QA)\text{ (mod }\pi).$

Hence four points $A,R,K,Q$ are cyclic. Therefore:

$\displaystyle(KB,KC)=(KR,KQ)=(AR,AQ)$ $\displaystyle=(AR,AY)+(AY,AQ)$
$\displaystyle=(BY,BC)+(CB,CY)$ $\displaystyle=(YB,YC)\text{ (mod }\pi).$

So four points $B,K,Y,C$ are cyclic. We have

$(KR,KY)=(KY,KC)=(BY,BC)=(AR,AY)\text{ (mod }\pi).$

Hence $Y$ is on $(ARK)\equiv(AQR)$. Moreover, by using the power of point’s
properties, we have the radical axis of $\gamma_{1}$ and $\gamma_{2}$ passes
their common tangents, following that $AY$ passes $BC$ at its midpoint or $AY$
is the median line of triangle $ABC$. The subsequent sections will go into the
various qualities associated with the aforementioned pair of fixed points,
which are closely linked to several notable triangle centers.

##  2\. Properties related to the first problem

Two above problems appeared in many contests [1, 2] and journals [4] in the
world, but it seems like there are just few properties about. This article
will put forward some results about two fixed points mentioned above. We
denotes $A_{X},A_{Y}$ be that two points in the previous problems.

######  Theorem 2.1.

$A_{X}$ is projection point of the circumcenter of triangle $ABC$ on the
$A-$symmedian line of that triangle.

Figure 3. Proof of Theorem 2.1

Proof. Let $A_{S}$ be the second intersection of $AA_{X}$ and $(ABC)$. From
Problem 1, we have two triangle $ABA_{X}$ and $AA_{X}C$ are homothetic hence
$A_{X}A^{2}=A_{X}B\cdot A_{X}C$. Moreover, we have

$(A_{X}B,A_{X}A_{S})=(A_{X}B,AA_{X})=(AA_{X},A_{X}C)=(A_{X}A_{S},A_{X}C),$
$\displaystyle(A_{S}B,A_{S}A_{X})=(CB,CA)$
$\displaystyle=(CB,CA_{X})+(CA_{X},CA)$
$\displaystyle=(CB,CA_{X})+(AA_{S},AB)$
$\displaystyle=(CB,CA_{X})+(CA_{S},CB)$ $\displaystyle=(CA_{S},CA_{X}).$

So two triangle $A_{X}BA_{S}$ and $A_{X}A_{S}C$ are homothetic. Therefore
$A_{X}A_{S}^{2}=A_{X}B\cdot A_{X}C$. Hence $A_{X}A=A_{X}A_{S}$ or $A_{X}$ is
midpoint of $AA_{S}$, following that $A_{X}$ is projection point of the
circumcenter of triangle $ABC$ on the $A$-symmedian line of that triangle.

######  Theorem 2.2.

Brocard circle [14] is the circle which passes two Brocard points
$Z_{1},Z_{2}$ [12, 13], Lemoine point $L$ [16] and circumcenter $O$ of
triangle $ABC$. Let $A_{X}$ be the fixed point mentioned in Problem 1. The
construction of $B_{X}$ and $C_{X}$ is performed in a similar manner, but with
the utilization of vertices $B$ and $C$. Then $A_{X}$, $B_{X}$ and $C_{X}$ lie
on Brocard circle [14] of triangle $ABC$.

######  Remark 2.1.

$A_{X}B_{X}C_{X}$ is the second Brocard triangle of triangle $ABC$.

######  Lemma 2.1.

(Mannheim’s theorem) Given triangle $ABC$ and three points $L$, $M$, $N$
respectively be on $BC$, $CA$, $AB$. Let $A^{\prime}$, $B^{\prime}$,
$C^{\prime}$ respectively be three points that lie on $(AMN)$, $(BNL)$,
$(CLM)$ such that $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$ are concurrent
at $K$. Prove that four points $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $K$
are cyclic.

Figure 4. Proof of Lemma 2.1

Proof. Let $P$ be Miquel point of triangle $LMN$ with respect to triangle
$ABC$. Let $Q$ be intersection of $PB^{\prime}$ and $(AMN)$. Applying Reim’s
theorem to $(AMN)$ and $(BNL)$ we have $AQ\parallel BB^{\prime}$. Through
angle chasing:

$(KA,KB^{\prime})=(AA^{\prime},AQ)=(PA^{\prime},PB^{\prime})\text{ (mod }\pi)$

Therefore $A^{\prime},P,B^{\prime},K$ are cyclic. Similarly,
$B^{\prime},P,C^{\prime},K$ are cyclic. Hence four points $A^{\prime}$,
$B^{\prime}$, $C^{\prime}$, $K$ are cyclic.

######  Lemma 2.2.

Given triangle $ABC$ inscribed in $(O)$ and a point $P$ is not this triangle’s
vertex. Lines $AP$, $BP$, $CP$ respectively cut $(ABC)$ at $S$, $T$, $U$. Let
$X$, $Y$, $Z$ be midpoints of segments $AS$, $BT$, $CU$, respectively. Prove
that $X$, $Y$, $Z$ lie on $(OP)$ (a circle with center $O$ and radius $OP$).

Proof. We have $X$, $Y$,$Z$ respectively be projection of $O$ on $AS$, $BT$,
$CU$. Through angle chasing:

$(XP,XO)=(YP,YO)=(ZP,ZO)=\frac{\pi}{2}.$

Therefore $X,Y,Z$ lie on $(OP)$. The lemma has been proved.

Back to Theorem 2.2,

Figure 5. Proof of Theorem 2.2

Proof. Applying Lemma 2.1, to triangle $ABC$ with $AA_{X},BB_{X},CC_{X}$ are
concurrent at $L$ and $Z_{1}$ is intersection of
$(ABB_{X}),(BCC_{X}),(CAA_{X})$ we have $L,Z_{1}$ lie on $(A_{X}B_{X}C_{X})$.
Similarly, $Z_{2}$ lies on $(A_{X}B_{X}C_{X})$. From Lemma 2.2, we have
$A_{X},B_{X},C_{X}$ lie on $OL$. The theorem has been proved.

######  Theorem 2.3.

Ray $AA_{Y}$ cuts $(ABC)$ at $A_{M}$. Prove that $BA_{Y}CA_{M}$ is
parallelogram.

Figure 6. Proof of Theorem 2.3

Proof. Using the properties of $A_{Y}$ at Problem 2, we have

$(BA_{Y},BC)=(AB,AA_{M})=(CB,CA_{M}).$

Therefore $A_{Y}B\parallel A_{M}C$. Similarly, $A_{Y}C\parallel A_{M}C$ so
$BA_{Y}CA_{M}$ is a parallelogram. The problem has been solved.

######  Corollary 2.1.

$A_{Y}$ is symmetric with $A_{S}$ through $BC$.

######  Theorem 2.4.

$A_{X}$ is the isogonal conjugate of $A_{Y}$ with respect to triangle $ABC$.

Figure 7. Proof of Theorem 2.4

Proof. Because $AA_{X}$ and $AA_{Y}$ respectively be symmedian line and median
line at vertex $A$ are two isogonal conjugate lines so we have to prove that
$BA_{X}$ and $BA_{Y}$ are two isogonal conjugate ones. Through angle chasing,

$(BA_{X},BA)=(AA_{X},AC)=(AB,AA_{Y})=(BC,BA_{Y}),$

we have $BA_{X}$ and $BA_{Y}$ are isogonal conjugate, therefore $A_{X}$ is the
isogonal conjugate of $A_{Y}$ with respect to triangle $ABC$.

######  Theorem 2.5.

Given triangle $ABC$, centroid $G$. Let $M_{AB}$, $M_{BC}$, $M_{CA}$
respectively be midpoints of $BC$, $CA$, $AB$. Let $A_{G}$ be midpoint of
$AA_{Y}$. Prove that $A_{G}$ lies on $(M_{AB}M_{BC}M_{CA})$.

Figure 8. Proof of Theorem 2.5

Proof. Because $M_{AB}A_{G}$ is the midline of triangle $ABA_{Y}$ so we have:

$(M_{AB}A,M_{AB}A_{G})=(BA,BA_{Y})$

Draw altitude $CH_{C}$ of triangle $ABC$. Through angle chasing,

$\displaystyle(M_{BC}H_{C},M_{BC}A)$ $\displaystyle=(M_{BC}H_{C},BA)+(BA,CB)$
$\displaystyle+(CB,CA)+(CA,M_{BC}A)$
$\displaystyle=2(BA,CB)+(CB,CA)+(CA,M_{BC}A)$
$\displaystyle=(BA,CB)+(AB,AM_{BC})$
$\displaystyle=(BA,BA_{Y})+(BA_{Y},CB)+(CB,CA_{M})$
$\displaystyle=(BA,BA_{Y})$ $\displaystyle=(M_{AB}A,M_{AB}A_{G}).$

Hence $A_{G}$ lies on $(M_{AB}H_{C}M_{BC})\equiv(M_{AB}M_{BC}M_{CA})$. We have
done.

######  Theorem 2.6.

Draw parallelogram $M_{AB}A_{G}M_{CA}A_{GY}$. Then $A_{GY}$ is midpoint of
$AA_{M}$.

Figure 9. Proof of Theorem 2.6

Proof. Let $A_{N}$ be midpoint of $M_{AB}M_{CA}$. Because
$M_{AB}A_{G}M_{CA}A_{GY}$ and $AM_{AB}M_{BC}M_{CA}$ are parallelogram, we can
easily see that $A_{N}$ is midpoint of both $A_{G}A_{GY}$ and $AM_{BC}$. We
have

$\displaystyle\overline{AA_{GY}}$
$\displaystyle=\overline{AA_{N}}+\overline{A_{N}A_{GY}}=\overline{A_{N}M_{BC}}+\overline{A_{G}A_{N}}$
$\displaystyle=\overline{A_{G}M_{BC}}=\overline{A_{G}A_{Y}}+\overline{A_{Y}M_{BC}}$
$\displaystyle=\frac{1}{2}\left(\overline{AA_{Y}}+\overline{A_{Y}A_{M}}\right)=\frac{1}{2}\overline{AA_{M}}.$

Therefore $A_{GY}$ is midpoint of $AA_{M}$.

######  Corollary 2.2.

$A_{X}A_{GY}\parallel BC$.

######  Corollary 2.3.

$A_{X}$ is symmetric with $A_{G}$ through $M_{AB}M_{CA}$.

######  Theorem 2.7.

$A_{Y}$ is the Anticomplement point of $A_{GY}$ with respect to triangle
$ABC$.

Proof. From these above properties, we have:

$\displaystyle\overline{GA_{GY}}$
$\displaystyle=\overline{GM_{BC}}-\overline{A_{GY}M_{BC}}=2\overline{A_{N}G}-\overline{A_{G}A_{Y}}$
$\displaystyle=\overline{2A_{N}G}-\overline{A_{G}A_{N}}-\overline{A_{N}A_{Y}}=\overline{A_{N}A_{Y}}+2\overline{A_{Y}G}-\overline{A_{N}A_{GY}}$
$\displaystyle=\overline{A_{GY}A_{Y}}+2\overline{A_{Y}G}=\overline{A_{GY}G}+\overline{A_{Y}G}.$

Hence $\overline{A_{Y}G}=2\overline{GA_{GY}}$ therefore $A_{Y}$ is the
Anticomplement point of $A_{GY}$ with respect to triangle $ABC$.

######  Theorem 2.8.

The first Brocard triangle is a triangle which has three vertices respectively
be intersection points of lines $AZ_{1}$, $BZ_{1}$, $CZ_{1}$, $AZ_{2}$,
$BZ_{2}$, $CZ_{2}$. Let $Z_{A},Z_{B},Z_{C}$ be three vertices of this
triangle. Prove that $Z_{A}Z_{B}Z_{C}$ is inscribed in $(A_{X}B_{X}C_{X})$.

Figure 10. Proof of Theorem 2.8

Proof. Let $(AZ_{1},AB)=(AC,AZ_{2})=\alpha$. Through angle chasing,

$\displaystyle(A_{X}Z_{1},A_{X}Z_{2})$
$\displaystyle=(A_{X}Z_{1},A_{X}C)+(A_{X}C,A_{X}A)+(A_{X}A,A_{X}Z_{2})$
$\displaystyle=(AZ_{1},AC)+(A_{X}A,A_{X}B)+(A_{X}A,Z_{2}A)+(Z_{2}A,A_{X}Z_{2})$
$\displaystyle=(AZ_{1},AC)+(A_{X}A,A_{X}B)+(A_{X}A,Z_{2}A)+(BA,BA_{X})$
$\displaystyle=(AZ_{1},AC)+(A_{X}A,A_{X}B)+(A_{X}A,Z_{2}A)+(AC,AA_{X})$
$\displaystyle=(AZ_{1},AB)+(AB,AA_{X})+(A_{X}A,A_{X}B)+(A_{X}A,Z_{2}A)$
$\displaystyle=(AZ_{1},AB)+(BA,A_{X}B)+(A_{X}A,Z_{2}A)$
$\displaystyle=(AZ_{1},AB)+(AC,AA_{X})+(AA_{X},AZ_{2})$
$\displaystyle=(AZ_{1},AB)+(AC,AZ_{2})$ $\displaystyle=2\alpha\text{ (mod
}\pi).$

On the other hand, we have

$(Z_{A}Z_{1},Z_{A}Z_{2})=(Z_{A}Z_{1},CB)+(CB,Z_{A}Z_{2})=\alpha+\alpha=2\alpha.$

Therefore $Z_{A}$ lies on $(Z_{1}A_{X}Z_{2})\equiv(A_{X}B_{X}C_{X})$.
Similarly, $Z_{B},Z_{C}$ lie on $(A_{X}B_{X}C_{X})$. WE have done.

######  Corollary 2.4.

$Z_{1}$ is symmetric with $Z_{2}$ through $OL$ and $(OZ_{1},OZ_{2})=2\alpha$.

######  Theorem 2.9.

$OA_{X}$, $CZ_{2}$ cut $\omega_{1}$ at $O_{W}$, $C_{W}$. Then
$O_{W}C_{W}\parallel OZ_{A}$.

######  Lemma 2.3.

Let $BZ_{1}$ cut $\omega_{2}$ at $B_{W}$. Prove that $B_{W}C_{W}\parallel BC$.

Proof. We have $(BB_{W},BC)=(CA,CZ_{1})=(B_{W}A,B_{W}Z_{1})$ so
$AB_{W}\parallel BC$. Similarly, $AC_{W}\parallel BC$ so $B_{W}C_{W}\parallel
BC$. We have done.

Back to Theorem 2.9,

Figure 11. Proof of Theorem 2.9

Proof. By Lemma 2.3 and $(BC,BZ_{A})=(CZ_{A},CB)$, we have $BCB_{W}C_{W}$ is
an isosceles trapezoid or $OZ_{A}$ is perpendicular to $B_{W}C_{W}$. On the
other hand, by Theorem 2.2, we have
$(A_{X}A,A_{X}O_{W})=(C_{W}A,C_{W}O_{W})=\frac{\pi}{2}$. Therefore
$O_{W}C_{W}$ is perpendicular to $B_{W}C_{W}$. So $O_{W}C_{W}\parallel
OZ_{A}$.

######  Theorem 2.10.

$A_{X}Z_{A}$, $B_{X}Z_{B}$, $C_{X}Z_{C}$ are concurrent at $G$.

######  Lemma 2.4.

Given triangle $ABC$. On the plane containing this triangle, choose
$A^{\prime}$, $B^{\prime}$, $C^{\prime}$ such that the triangles
$A^{\prime}CB$, $B^{\prime}AC$, $C^{\prime}BA$ are homothetic. We construct
the parallelogram $BA^{\prime}CD$. Prove that $A^{\prime}CDB$ is also a
parallelogram.

Figure 12. Proof of Lemma 2.4

Proof. We have

$\frac{C^{\prime}B}{C^{\prime}D}=\frac{C^{\prime}B}{AB^{\prime}}=\frac{C^{\prime}A}{B^{\prime}C}=\frac{B^{\prime}D}{B^{\prime}C}$

and

$\displaystyle(C^{\prime}B,C^{\prime}D)$
$\displaystyle=(C^{\prime}B,C^{\prime}A)+(C^{\prime}A,C^{\prime}D)$
$\displaystyle=(B^{\prime}A,B^{\prime}C)+(B^{\prime}D,B^{\prime}A)=(B^{\prime}D,B^{\prime}C).$

Therefore triangle $C^{\prime}BD$ and triangle $B^{\prime}DC$ are homothetic.
Hence

$\frac{BD}{CD}=\frac{C^{\prime}B}{B^{\prime}D}=\frac{C^{\prime}B}{C^{\prime}A}=\frac{A^{\prime}B}{A^{\prime}C}.$

On the other hand,

$\displaystyle(DB,DC)$
$\displaystyle=(DB,DC^{\prime})+(DC^{\prime},DB^{\prime})+(DB^{\prime},DC)$
$\displaystyle=(DC,B^{\prime}C)+(B^{\prime}A,DB^{\prime})+(DB^{\prime},DC)$
$\displaystyle=(B^{\prime}A,B^{\prime}C)$ $\displaystyle=(A^{\prime}C,AB).$

So triangle $DBC$ and triangle $A^{\prime}CB$ are homothetic. Therefore,
$BD\parallel A^{\prime}C$ and $CD\parallel A^{\prime}B$. We conclude that
$A^{\prime}CDB$ is a parallelogram.

######  Lemma 2.5.

$G$ is the centroid of triangle $Z_{A}Z_{B}Z_{C}$.

Figure 13. Proof of Lemma 2.5

Proof. Let $Z^{\prime}_{A}$ be symmetric point with $Z_{A}$ through $BC$.
Since triangle $BZ_{A}C$ is a isosceles triangle at $Z_{A}$,
$BZ_{A}CZ^{\prime}_{A}$ is a parallelogram. Applying Lemma 2.4 for triangle
$ABC$, we have $AZ_{C}Z^{\prime}_{A}Z_{B}$ is a parallelogram so
$AZ^{\prime}_{A}$ passes midpoint $N_{BC}$ of $Z_{B}Z_{C}$. In triangle
$AZ_{A}Z^{\prime}_{A}$, the two median lines $AM_{BC}$ and $Z_{A}N_{BC}$
intersect with each other at this triangle’s centroid so that this point is
also the centroid $G$ of triangle $ABC$ and triangle $Z_{A}Z_{B}Z_{C}$.

######  Lemma 2.6.

Given triangle $ABC$ and two disjoint arbitrary points $P$, $Q$. Let $X$, $Y$,
$Z$ respectively be projection of $P$ on the lines which passes $Q$ and is
perpendicular to $BC$, $CA$, $AB$. Let $X^{\prime}$, $Y^{\prime}$,
$Z^{\prime}$ respectively be projection of $Q$ on $AP$, $BP$, $CP$. Prove that
$XX^{\prime}$, $YY^{\prime}$ and $ZZ^{\prime}$ are concurrent.

Figure 14. Proof of Lemma 2.6

Proof. Noted that $X$, $Y$, $Z$, $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$ lie
on a circle with diameter $PQ$ and $AB\parallel PZ$, $AC\parallel PY$ so we
have

$(AB,AC)=(ZP,PY)=(XY,XZ).$

Similarly, $(BC,BA)=(YZ,YX)$ and $(CA,CB)=(ZX,ZY)$ so two triangles $XYZ$ and
$ABC$ are homothetic. Let $P^{\prime}$ be the image of $P$ through the
homothety transformation which turns triangle $ABC$ into triangle $XYZ$. Let
$P^{*}$ be the isogonal conjugate point of $P$ in triangle $XYZ$. We have

$(XX^{\prime},XZ)=(PX^{\prime},PZ)=(AP,AB)=(XP^{*},XZ).$

Therefore $XX^{\prime}$ passes $P^{*}$. Similarly, $YY^{\prime}$ and
$ZZ^{\prime}$ passes $P^{*}$, which solves the lemma.

Figure 15. Proof of Theorem 2.10

Back to Theorem 2.10,

Proof. By applying Lemmas 2.5 and 2.6 for triangle $Z_{A}Z_{B}Z_{C}$, we
solves the problem.

By Corollary 2.3, the Brocard circle is definitely the Hagge circle [5] of $G$
in triangle $M_{BC}M_{CA}M_{AB}$. We will discuss some properties related to
this issue and use these ones to prove Theorem 2.10.

######  Lemma 2.7 (Hagge circle).

Given triangle $ABC$, orthocenter $H$ and a point $P$ lies inside $(ABC)$.
Rays $AP$, $BP$, $CP$ cuts $(ABC)$ at the second points $A_{1}$, $B_{1}$,
$C_{1}$. Let $A_{2}$, $B_{2}$, $C_{2}$ respectively be the symmetry points of
$A_{1}$, $B_{1}$, $C_{1}$ through $A$, $B$, $C$. Prove that $A_{2}$, $B_{2}$,
$C_{2}$ and $H$ are cyclic.

######  Remark 2.2.

The circle mentioned above called Hagge circle $\mathcal{H}_{P}$ of point $P$
with respect to triangle $ABC$.

Figure 16. Proof of Lemma 2.7

Proof. Let $P^{\prime}$ be the isogonal conjugate of $P$ in triangle $ABC$.
$AP^{\prime}$, $BP^{\prime}$, $CP^{\prime}$ cut $(ABC)$ respectively at
$A^{\prime}_{1}$, $B^{\prime}_{1}$, $C^{\prime}_{1}$. Let $B^{\prime}_{2}$,
$C^{\prime}_{2}$ respectively be symmetry points of $B_{2}$, $C_{2}$ through
midpoint of $BC$. Set $O$ is the center of $(ABC)$ with diameter
$AA^{\prime}$. It is easy to see that $A_{2}$, $B_{2}$, $C_{2}$ are symmetry
points of $A^{\prime}_{1}$, $B^{\prime}_{2}$, $C^{\prime}_{2}$ through
midpoint of $BC$. We have $CB_{2}AB^{\prime}_{1}$ and $CB_{2}BB^{\prime}_{2}$
are parallelograms so the quadrilateral $ABB^{\prime}_{2}B^{\prime}_{1}$ is
also a parallelogram. We conclude that midpoint $M_{1}$ of $AB^{\prime}_{2}$
is the projection of $O$ on $BP^{\prime}$. Similarly, midpoint $M_{2}$ of
$AC^{\prime}_{2}$ is the projection of $O$ on $CP^{\prime}$. Let $M_{3}$ be
midpoint of $AA^{\prime}_{1}$. Then triangle $M_{1}M_{2}M_{3}$ is inscribed in
a circle with diameter $OP^{\prime}$. We use two geometric transformations
respectively with the first one (1) is the homothety transformation
$\mathbf{H}_{A}^{2}$ with center $A$ and ratio 2; and the later (2) is the
symmetric one $\mathbf{S}_{M_{BC}}$ through midpoint $M_{BC}$ of $BC$, to
points $M_{3}$, $M_{1}$, $M_{2}$, $O$ such that

$\mathbf{S}_{M_{BC}}\circ\mathbf{H}_{A}^{2}\colon M_{3},M_{1},M_{2},O\mapsto
A_{2},B_{2},C_{2},H.$

Since $M_{1},M_{2},M_{3},O$ lie on a same circle so $A_{2},B_{2},C_{2},H$ are
also cyclic.

######  Lemma 2.8.

Given triangle $ABC$ and two points $S$ and $S^{\prime}$ are isogonal
conjugate. Let $X$, $X^{\prime}$ respectively be intersection of $AS$,
$AS^{\prime}$ with $(ABC)$. Let $V$ be intersection of $AS^{\prime}$ and $BC$.
Prove that

$\frac{\overline{AS}}{\overline{SX}}=\frac{\overline{S^{\prime}V}}{\overline{VX^{\prime}}}.$

Figure 17. Proof of Lemma 2.8

Proof. We have

$(CV,CX^{\prime})=(AB,AX^{\prime})=(AX,AC),$
$(X^{\prime}V,X^{\prime}C)=(XA,XC)$

So two triangles $VCX^{\prime}$ and $CAX$ are homothetic. On the other hand,

$\displaystyle(S^{\prime}X^{\prime},S^{\prime}C)$
$\displaystyle=(AS^{\prime},AC)+(CA,CS^{\prime})$
$\displaystyle=(AB,AS)+(CS,CB)$ $\displaystyle=(CB,CX)+(CS,CB)$
$\displaystyle=(CS,CX),$ $(X^{\prime}S^{\prime},X^{\prime}C)=(XS,XC).$

Therefore two triangles $CX^{\prime}S^{\prime}$ and $SXC$ are homothetic. So,

$\overline{X^{\prime}V}\cdot\overline{XA}=\overline{X^{\prime}C}\cdot\overline{XC}=\overline{X^{\prime}S^{\prime}}\cdot\overline{XS}.$

Hence,
$\frac{\overline{XA}}{\overline{XS}}=\frac{\overline{X^{\prime}S^{\prime}}}{\overline{X^{\prime}V}}$
or
$\frac{\overline{AS}}{\overline{SX}}=\frac{\overline{S^{\prime}V}}{\overline{VX^{\prime}}}$.

######  Lemma 2.9.

Let $A_{3}$, $B_{3}$, $C_{3}$ respectively be intersection of
$\mathcal{H}_{P}$ and $AH$, $BH$, $CH$. Prove that $A_{2}A_{3}$, $B_{2}B_{3}$,
$C_{2}C_{3}$ are concurrent at $P$.

Figure 18. Proof of Lemma 2.9

Proof. We construct diameter $HH^{\prime}$ of $\mathcal{H}_{P}$. From Lemma
2.7, we have

$\mathbf{S}_{M_{BC}}\circ\mathbf{H}_{A}^{2}\colon O,P^{\prime}\mapsto
H,H^{\prime}.$

Therefore $OP^{\prime}\parallel HH^{\prime}$ and
$\overline{H^{\prime}H}=2\overline{OP^{\prime}}$ so $H^{\prime}$ is the
Anticomplement point of $P^{\prime}$ in triangle $ABC$. On the other hand,
$A_{3}H^{\prime}\parallel BC$ and $BC\perp AH$ hence
$AA^{\prime}=2\text{d}(P^{\prime},BC)$. Using Lemma 2.8, we have

$\frac{\overline{AP}}{\overline{PA_{1}}}=\frac{\text{d}(P^{\prime},BC)}{\text{d}(A^{\prime}_{1},BC)}=\frac{\text{d}(P^{\prime},BC)}{\text{d}(A_{1},BC)}=\frac{\overline{AA_{3}}}{\overline{A_{2}A_{1}}}.$

This means $A_{2}A_{3}$ passes $P$. Similarly, $B_{2}B_{3},C_{2}C_{3}$ passes
$P$. We have solved the lemma.

Back to Theorem 2.10,

Figure 19. Proof of Theorem 2.10

Proof. By Corollary 2.3, the Brocard cirlce is the Hagge circle of $G$ with
respect to triangle $M_{BC}M_{CA}M_{AB}$. By applying Lemma 2.7 and 2.9 for
triangle $M_{BC}M_{CA}M_{AB}$, we have the result.

##  3\. Properties related to the second problem

######  Theorem 3.1.

$A_{Y}$ is the projection of orthocenter $H$ on the $A-$median line of
triangle $ABC$.

Figure 20. Proof of Theorem 3.1

Proof. Let $AA_{O}$ be diameter of $(ABC)$. Since $BHCA_{O}$ is a
parallelogram, $M_{BC}$ is the midpoint of $A_{O}H$. On the other hand,
$M_{BC}$ is also the mid point of $A_{Y}A_{M}$. By Theorem 2.3,
$HA_{Y}A_{O}A_{M}$ is a parallelogram therefore $A_{Y}H\parallel A_{O}A_{M}$.
Moreover, $A_{O}A_{M}\perp AA_{M}$ hence $A_{Y}$ is the projection of $H$ on
the $A$-median line.

######  Corollary 3.1.

Let $B_{Y}$ and $C_{Y}$ be constructed in a similar manner as $A_{Y}$, but
with respect to $B$ and $C$ respectively. Let $H_{A}H_{B}H_{C}$ be the
Orthocentroidal triangle with $H_{A}$, $H_{B}$, $H_{C}$ repsectively be
projection of $G$ on $AH$, $BH$, $CH$. Then two triangles $A_{Y}B_{Y}C_{Y}$
and $H_{A}H_{B}H_{C}$ are inscribed in the circle with diameter $GH$ which
called Orthocentroidal circle.

######  Corollary 3.2.

The Hagge circle of the Lemoine point $L$ in triangle $ABC$ is the
Orthocentroidal circle of this triangle.

Figure 21. Demonstration of Corollary 3.2

######  Theorem 3.2.

The radical line of $\left(A_{X}B_{X}C_{X}\right)$ and
$\left(A_{Y}B_{Y}C_{Y}\right)$ passes midpoint $X\left(39\right)$ of two
Brocard points.

Figure 22. Demonstration of Theorem 3.2

We can see the proof of this property here [9] using Barycentric coordinate.

######  Theorem 3.3.

$A_{Y}H_{A}$, $B_{Y}H_{B}$, $C_{Y}H_{C}$ are concurrent at $L$.

######  Lemma 3.1.

$L$ is also the Lemoine point of triangle $H_{A}H_{B}H_{C}$.

Proof. By Corollary 3.2, the Orthocentroidal circle of triangle $ABC$ is the
Hagge circle of $L$ with respect to this triangle. Moreover, by Corollary 2.1,
triangle $A_{Y}B_{Y}C_{Y}$ is the Hagge triangle of $L$, which means $A_{Y}$,
$B_{Y}$, $C_{Y}$ are constructed in the same way of $A_{2}$, $B_{2}$, $C_{2}$
in Lemma 2.7, but with respect to $L$. We then apply Lemma 2.9 to completely
solve this result.

Back to Theorem 3.3,

Proof. By applying Lemma 2.7, 2.9 and 3.1 for triangle $ABC$ with Lemoine
point $L$ and the Orthocentroidal circle, we can prove the theorem.

Figure 23. Proof of Theorem 3.3

######  Theorem 3.4.

The perpendicular bisector of $AXA_{GY}$ cuts $AL$ at $L_{N}$. Then
$AXA_{Y}\parallel L_{N}M_{BC}$.

Figure 24. Proof of Theorem 3.4

Proof. By Corollary 2.2, $L_{N}$ also lies on the perpendicular bisector of
$M_{AB}M_{CA}$ containing $A_{N}$ so $A_{N}L_{N}\parallel A_{Y}A_{S}$.
Therefore

$\overline{AL_{N}}\cdot\overline{AA_{Y}}=\overline{AA_{N}}\cdot\overline{AA_{S}}=\overline{AM_{BC}}\cdot\overline{AA_{X}}$

so
$\frac{\overline{AA_{X}}}{\overline{AL_{N}}}=\frac{\overline{AA_{Y}}}{\overline{AM_{BC}}}$.
Hence, $A_{X}A_{Y}\parallel L_{N}M_{BC}$.

######  Theorem 3.5.

$\left(A_{X}A_{GY}L_{N}\right)$ is tangent to $\left(BOC\right)$.

Figure 25. Proof of Theorem 3.5

Proof. By Theorem 2.1, $A_{X}$ lies on $(BOC)$. Let $Ad$ be a ray which
belongs to tangent line at $A$ of $\left(A_{X}A_{GY}L_{N}\right)$ such that
$Ad$ and $B$ are on the same side of the plane separated by $AL$. By Theorem
2.1, $A_{X}A_{S}$ is the $A_{X}$-bisector of triangle $A_{X}BC$. We have

$\displaystyle(A_{X}d,A_{X}B)$
$\displaystyle=(A_{X}d,A_{X}A_{S})+(A_{X}A_{S},A_{X}B)$
$\displaystyle=(A_{X}A_{S},A_{X}A_{GY})+(A_{X}C,A_{X}A_{S})$
$\displaystyle=(A_{X}C,A_{X}A_{GY})$ $\displaystyle=(CB,CA_{X}).$

So $Ad$ is the tangent line of $(A_{X}BC)\equiv(BOC)$. We have done.

######  Theorem 3.6.

Let $A_{LX}$ be projection of $H$ on $AL$. Then $M_{BC}A_{Y}=M_{BC}A_{LX}$.

Figure 26. Proof of Theorem 3.6

Proof. Draw altitudes $BH_{CA}$ and $CH_{AB}$ of triangle $ABC$. It can be
seen that $A,H,H_{CA},H_{AB},A_{Y},A_{LX}$ lie on the circle with diameter
$AH$. Since $M_{BC}$ is the intersection of two tangent lines at $H_{CA}$ and
$H_{AB}$ of $\left(AH\right)$ so $AH_{AB}A_{Y}H_{BC}$ is a harmonic
quadrilateral. Since $AL$ is a median line of triangle $AH_{CA}H_{AB}$ hence
$H_{CA}H_{AB}\parallel A_{Y}A_{LX}$. Moreover, as $M$ lies on the
perpendicular bisector of $H_{CA}H_{AB}$, which is also the perpendicular
bisector of $A_{Y}A_{LX}$, so that $M_{BC}A_{Y}=M_{BC}A_{LX}$.

######  Corollary 3.3.

$\left(A_{Y}M_{BC}A_{LX}\right)$ is tangent to the Euler circle of triangle
$ABC$.

######  Theorem 3.7.

Let $L_{BC}$ be intersection of $AL$ and $BC$. Then $A_{Y}L_{BC}$ is a
symmedian line of triangle $A_{Y}BC$.

Figure 27. Proof of Theorem 3.7

Proof. Since $AL_{BC}$ is a symmedian line of triangle $ABC$ so

$\frac{\overline{L_{BC}B}}{\overline{L_{BC}C}}=-\left(\frac{AB}{AC}\right)^{2}.$

On the other hand,

$\left(\frac{AB}{AC}\right)^{2}=\left(\frac{AB}{M_{BC}B}\cdot\frac{M_{BC}C}{AC}\right)^{2}=\left(\frac{A_{M}C}{A_{M}M_{BC}}\cdot\frac{A_{M}M_{BC}}{A_{M}B}\right)^{2}=\left(\frac{A_{Y}B}{A_{Y}C}\right)^{2}$

So

$\frac{\overline{L_{BC}B}}{\overline{L_{BC}C}}=-\left(\frac{A_{Y}B}{A_{Y}C}\right)^{2}.$

We conclude that $A_{Y}L_{BC}$ is a symmedian line of triangle $A_{Y}BC$.

######  Theorem 3.8.

$\left(A_{Y}M_{BC}A_{LX}\right)$ is tangent to $\left(BHC\right)$.

######  Lemma 3.2.

Given triangle $ABC$. On segment $BC$, we choose $D$, $E$ such that $AD$, $AE$
are isogonal conjugate of each other in this triangle. Prove that
$\left(ABC\right)$ is tangent to $\left(ADE\right)$.

Figure 28. Proof of Lemma 3.2

Proof. The tangent line at $A$ of $\left(ABC\right)$ cuts line $BC$ at $F$.
Then

$(AF,AD)=(AF,AB)+(AB,AD)=(CA,CB)+(EA,CA)=(EA,CB)$

so $AF$ is tangent to $\left(ADE\right)$. Thus, $\left(ABC\right)$ is tangent
to $\left(ADE\right)$.

Back to Theorem 3.8,

Figure 29. Proof of Theorem 3.8

Proof. Draw the altitude $AH_{BC}$ of triangle $ABC$. We have that $H_{BC}$
and $A_{LX}$ lie on the circle with diameter $HL_{BC}$. Therefore,

$(L_{BC}A_{LX},L_{BC}M_{BC})=(HA_{LX},HH_{BC})=(A_{Y}A_{LX},A_{Y}A).$

So $A_{Y},A_{LX},L_{BC},M_{BC}$ are cyclic. Using Theorem 3.7, $A_{Y}L_{BC}$
is a symmedian line of triangle $A_{Y}BC$. We use Lemma 3.2 to conclude that
$\left(A_{Y}M_{BC}A_{LX}\right)$ touches $\left(BHC\right)$.

######  Theorem 3.9.

The rays $BA_{Y}$, $CA_{Y}$ cuts $\left(ABC\right)$ at $A_{BY}$, $A_{CY}$. Let
$A^{\prime}_{BY},A^{\prime}_{CY}$ be symmetry points of $A_{BY},A_{CY}$
through $CA$, $AB$ respectively. Then $A^{\prime}_{BY}$ and $A^{\prime}_{CY}$
lie on $AL$.

Figure 30. Proof of Theorem 3.9

Proof. We have

$\displaystyle(AA^{\prime}_{BY},AC)$ $\displaystyle=(AC,AA_{BY})=(BC,BA_{BY})$
$\displaystyle=(CB,CA_{M})=(AB,AA_{M})$ $\displaystyle=(AL,AC).$

Thus $AL$ passes $A^{\prime}_{BY}$. Similarly, $AL$ passes $A^{\prime}_{CY}$.

######  Theorem 3.10.

$A_{LX}$ is midpoint of $A^{\prime}_{BY}A^{\prime}_{CY}$.

Proof. Rays $BH$ and $CH$ cut $\left(ABC\right)$ respectively at $H_{B}$ and
$H_{C}$. We have

$(H_{B}A_{CY},H_{B}B)=(CA_{CY},CB)=(HA_{Y},HH_{B})$

so $H_{B}A_{CY}\parallel HA_{Y}$, where $HA_{Y}\perp AG$. Simiarly
$H_{C}A_{BY}\parallel HA_{Y}$, where $HA_{Y}\perp AG$ hence
$H_{B}A_{CY}H_{C}A_{BY}$ is a isosceles trapezoid. We conclude that
$H_{C}A_{CY}=H_{B}A_{BY}$. Using symmetry transformation through $AB$ and
$AC$, we have $\mathbf{S}_{AB}\colon H,A_{CY}\mapsto H_{C},A^{\prime}_{CY}$
and $\mathbf{S}_{AC}\colon H,A_{BY}\mapsto H_{B},A^{\prime}_{BY}$. Therefore
$HA^{\prime}_{BY}=H_{B}A_{BY}=H_{C}A_{CY}=HA^{\prime}_{CY}$, which means
triangle $HA^{\prime}_{BY}A^{\prime}_{CY}$ is isosceles at $H$. Moreover,
$A_{LX}$ is the projection of $H$ on $A^{\prime}_{BY}A^{\prime}_{CY}$ so
$A_{LX}$ is midpoint of $A^{\prime}_{BY}A^{\prime}_{CY}$.

##  4\. Conclusion

This article is a discussion on the properties associated with a pair of
distinct triangle centers. Further investigations can be conducted in this
domain to delve deeper into the subject matter and uncover additional
distinctive characteristics pertaining to triangles.

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FACULTY OF MATHEMATICS AND COMPUTER SCIENCE, UNIVERSITY OF SCIENCE, VIETNAM
NATIONAL UNIVERSITY HO CHI MINH CITY, 227 NGUYEN VAN CU, WARD 4, DISTRICT 5,
HO CHI MINH CITY

E-mail address<EMAIL_ADDRESS>