# Inscribed triangles of Jordan curves in $\mathbb{R}^{n}$

Aryaman Gupta  Euler Circle, Mountain View, CA 94040<EMAIL_ADDRESS>and  Simon Rubinstein-Salzedo  Euler Circle, Mountain View, CA 94040
<EMAIL_ADDRESS>
###### Abstract.

Nielsen’s theorem states that any triangle can be inscribed in a planar Jordan
curve. We prove a generalisation of this theorem, extending to any Jordan
curve $J$ embedded in $\mathbb{R}^{n}$, for a restricted set of triangles. We
then conclude by investigating a condition under which a given point of $J$
inscribes an equilateral triangle in particular.

##  1\. Introduction

A _Jordan curve_ is a continuous image of the unit interval in
$\mathbb{R}^{n}$ that is injective everywhere except the endpoints, which are
mapped to the same point. A polygon is _inscribed_ in a Jordan curve if the
vertices of the polygon lie on the curve. There has been a considerable amount
of interest surrounding the inscription of triangles and quadrilaterals in
Jordan curves embedded in the plane. A lot of this interest stems from the
Toeplitz square peg conjecture, which asks whether any Jordan curve in the
plane has an inscribed square. Detailed exposition on this and similar
problems can be found in [Mat14] and [Pak10].

In the literature, some of these variants have already been resolved. See, for
example, [Mey81] and [Nie92], wherein it is shown, respectively, that a planar
Jordan curve necessarily inscribes a rectangle and any particular triangle.
Yet, the original conjecture itself remains unproven, except under certain
geometric or topological conditions. See, for instance, [Str89], [Mat14],
[Emc16] and [NW95].

Here, instead of squares, we shall consider the inscription of triangles,
under geometric conditions on the curve. The motivation for this paper comes
from the following two results, proven respectively in in [Nie92] and [Mey80].

######  Theorem 1.1 (Nielsen).

Let $J\subset\mathbb{R}^{2}$ be a Jordan curve and let $\triangle$ be any
triangle. Then infinitely many triangles similar to $\triangle$ can be
inscribed in $J$.

######  Theorem 1.2 (Meyerson).

Let $J\subset\mathbb{R}^{2}$ be a Jordan curve. For every point $p\in J$
except at most two, there exists an inscribed equilateral triangle such that
one of its vertices is $p$.

There has also been a smaller, but still significant, amount of interest in
higher dimensional variants; see for instance [Str89], [Mak16], [AK13], and
[NW95]. There are at least two difficulties in dealing with Jordan curves in
at least three dimensions. Firstly, many proofs regarding inscription for
planar Jordan curves (including the proof of Theorem 1.1) rely upon the
property that $J$ divides the plane into two disconnected subsets. Since this
does not, of course, generalise to higher dimensions, these proofs cannot be
generalised in any obvious way. Secondly, since Jordan curves in higher
dimensions are able to form knots, they can potentially be much more
pathological than planar Jordan curves.

The aim of this paper is to prove that, subject to a certain geometric
restriction, any triangle can be inscribed in a given Jordan curve $J$
embedded in $\mathbb{R}^{n}$.

Before stating the main result, we introduce some notation.

  * •

$J$ denotes a Jordan curve embedded in $\mathbb{R}^{n}$, defined by
$\gamma:[0,1]\rightarrow\mathbb{R}^{n}$.

  * •

Let $\delta\in(0,\frac{1}{2})$. Then,
$\Theta_{\delta}:(0,\delta)\times(0,\delta)\rightarrow\mathbb{R}_{\geq 0}$
denotes the function that maps each pair
$(s,s^{\prime})\in(0,\delta)\times(0,\delta)$ to the angle between
$\vv{o\gamma(s)}$ and $\overrightarrow{o\gamma(s^{\prime})}$.

  * •

$\Theta^{\prime}_{\delta}:(1-\delta,1)\times(0,\delta)\rightarrow\mathbb{R}_{\geq
0}$ denotes the function that maps each pair
$(s,s^{\prime})\in(1-\delta,1)\times(0,\delta)$ to the angle between
$\overrightarrow{o\gamma(s)}$ and $\overrightarrow{o\gamma(s^{\prime})}$.

  * •

$\triangle abc$ denotes the triangle in $\mathbb{R}^{n}$ with vertices
$a,b,c\in J$.

Here is our main result.

######  Theorem 1.3.

Let $\theta_{v}$ be the angle of some vertex $v$ of $\triangle$. If there
exists a $\theta_{v}$ such that

${\limsup_{\delta\rightarrow
0^{+}}\Theta_{\delta}<\theta_{v}<\liminf_{\delta\rightarrow
0^{+}}\Theta^{\prime}_{\delta}},$

then there exist two points $p,q\in J\setminus\\{o\\}$ such that $\triangle
opq$ is similar to $\triangle$, with vertex $v$ corresponding to $o$.

Note that the hypothesis of Theorem 1.3 is automatically satisfied when $o$ is
a smooth point.

For the proof, our first concern is how we identify when $o,p,q\in J$ inscribe
$\triangle$. Assuming $\triangle opq$ is similar to $\triangle$, let
$r\colonequals\frac{\|q-o\|}{\|p-o\|}$ and
$r^{\prime}\colonequals\frac{\|q-p\|}{\|p-o\|}$, where $r\geq 1$ without loss
of generality.

We fix $o$ to be $\gamma(0)$ and assume that it is the origin of
$\mathbb{R}^{n}$, and we let $p$ be any element of $J-\\{o\\}$. Then, the set
$\operatorname{tri}(o,p)$ of points $q$ such that $\triangle opq$ is similar
to $\triangle$ is an $(n-2)$-sphere. Formally,

$\operatorname{tri}(o,p)\colonequals\left\\{q\in\mathbb{R}^{n}\,\middle|\,\frac{\|\overrightarrow{Q}\|}{\|\overrightarrow{P}\|}=r\right\\}\cap\left\\{q\in\mathbb{R}^{n}\,\middle|\,\frac{\|\overrightarrow{Q}-\overrightarrow{P}\|}{\|\overrightarrow{P}\|}=r^{\prime}\right\\},$

where $\overrightarrow{Q}\colonequals\overrightarrow{oq}$ and
$\overrightarrow{P}\colonequals\overrightarrow{op}$. Since both sets of the
intersection are $(n-1)$-spheres centred at $o$ and $p$ respectively,
$\operatorname{tri}(o,p)$ is an $(n-2)$-sphere. See Figure 1 for a diagram
that illustrates this.

1.000000 -1.000000 2 -2$\overrightarrow{P}$2 -2$\overrightarrow{Q_{1}}$2
-2$\overrightarrow{Q_{2}}$2 -2$\overrightarrow{Q_{1}}$ \-
$\overrightarrow{P}$2 -2$\overrightarrow{Q_{2}}$ \- $\overrightarrow{P}$2
-2$tri(o,p)=\\{q_{1},q_{2}\\}$2 -2$tri(o,p)=C$2 -2$C$2
-2$\overrightarrow{Q_{3}}$2 -2$\overrightarrow{Q_{2}}$2
-2$\overrightarrow{P}$2 -2$\mathbb{R}^{2}$2 -2$\mathbb{R}^{3}$2 -22 -22
-2$\overrightarrow{Q_{2}}-\overrightarrow{P}$2
-2$\overrightarrow{Q_{3}}-\overrightarrow{P}$2 -2$\overrightarrow{Q_{4}}$2
-2$\overrightarrow{Q_{1}}$2 -2$Q_{2}$2 -2$Q_{1}$2 -2$p$2 -2$Q_{2}$2 -2$p$2 -22
-2$Q_{3}$

Figure 1. The set $\operatorname{tri}(o,p)$ in $\mathbb{R}^{2}$ and
$\mathbb{R}^{3}$ respectively. For the former space, this set is the 0-sphere
$\\{q_{1},q_{2}\\}$. For the latter space, this set is a a 1-sphere.

In order to prove Theorem 1.3, it suffices to show that there exists some
$p\in J-\\{o\\}$ such that $J\cap\operatorname{tri}(o,p)\neq\varnothing$.

Here is a more detailed outline of the path we take to prove our main result.

  * •

Setup: In §2.1, we introduce for $\operatorname{tri}(o,\gamma(t))\colonequals
S_{t}$, a scaled isometry $I_{t}$ such that $I_{t}(S_{t})$ is equal to the
same $(n-2)$-sphere $S^{n-2}_{\operatorname{can}}$ for all $t\in(0,1)$. This
isometry reorganises our coordinate system so that $S_{t}$ is mapped to a
constant frame of reference for any $t\in(0,1)$. Since we consider the
complement of $S_{t}$ in $\mathbb{R}^{n}$, it will be more convenient to
reorganise our coordinate system to make $S_{t}$ appear stationary, rather
than considering a moving complement. In §2.2, we assume that $\gamma$ and
$S_{t}$ never intersect. Under this assumption, we prove that
$I_{t}(\gamma)\simeq I_{t^{\prime}}(\gamma)$ in
$X\colonequals\mathbb{R}^{n}-S^{n-2}_{\operatorname{can}}$ for all
$t,t^{\prime}\in(0,1)$, where $\simeq$ denotes that two loops are freely
homotopic in $X$.

  * •

Finding $t_{1}$: In §3 we prove Lemma 3.2, which states that
$\gamma^{\prime}_{t_{1}}\colonequals I_{t_{1}}(\gamma)$ is homotopic in $X$ to
the trivial loop $c$ at $I_{t_{1}}(o)$ for some $t_{1}\in(0,1)$.

  * •

Finding $t_{2}$: In §4, we prove a series of technical lemmata (namely,
Lemmata 4.2 to 4.15) leading to a proof of Lemma 4.1, which states that
$\gamma^{\prime}_{t_{2}}$ is not homotopic in $X$ to $c$ for some
$t_{2}\in(0,1)$. We combine Lemmata 3.2 and 4.1 to show that
$\gamma^{\prime}_{t_{1}}\not\simeq\gamma^{\prime}_{t_{2}}$ if $\gamma\cap
S_{t}=\varnothing$ for every $t\in(0,1)$. However, from Lemma 2.1, we also
know that $\gamma^{\prime}_{t_{1}}\simeq\gamma^{\prime}_{t_{2}}$ if
$\gamma\cap S_{t}=\varnothing$ for every $t\in(0,1)$. Since both of these
results follow from the same hypothesis—namely, that $\gamma\cap
S_{t}=\varnothing$ for every $t\in(0,1)$—we know then that this hypothesis is
false. We then prove Theorem 1.3 and an additional corollary.

  * •

Inscribing equilateral triangles: In §5 we generalise Theorem 1.2 by showing
that any point $o\in J$ inscribes an equilateral triangle if $J$ satisfies a
certain condition.

See Figure 2 for diagrams outlining the sketch of the proof.

1.000000 -1.000000 (22.689000,5.391710)
(36.039000,3.841710)(22.764000,20.521700)(22.764000,20.421700)
(22.764000,20.321700)(9.338950,6.941710)(22.689000,5.391710)
(17.650000,17.250000)
(19.000000,17.400000)(21.800000,14.850000)(20.450000,14.800000)
(19.100000,14.750000)(16.300000,17.100000)(17.650000,17.250000) 2.2 -2.2$o$3
-3$S_{t_{2}}$2.2 -2.2$\gamma(t_{2})$

(a) At $t_{2}$, $S_{t_{2}}$ has not yet crossed through $\gamma$. In turn,
$\gamma^{\prime}_{t_{2}}$ is not homotopic in $X$ to $c$.

1.000000 -1.000000 3 -3$S_{T}$3 -33 -3$o$3 -33 -3$\gamma$3 -33
-3$\gamma(T)$(23.075000,5.530000)
(36.425000,3.980000)(23.150000,20.660000)(23.150000,20.560000)
(23.150000,20.460000)(9.725000,7.080000)(23.075000,5.530000)
(9.250000,15.000000)
(19.100000,15.050000)(35.000000,6.050000)(25.550000,6.050000)
(16.100000,6.050000)(-0.600000,14.950000)(9.250000,15.000000)

(b) At $T$ exactly, $S_{T}$ crosses through $\gamma$.

1.000000 -1.000000 (23.225000,5.680000)
(36.575000,4.130000)(23.300000,20.810000)(23.300000,20.710000)
(23.300000,20.610000)(9.875000,7.230000)(23.225000,5.680000)
[fillstyle=solid,fillcolor=diafillcolor,linecolor=diafillcolor]
(23.225000,5.680000)
(36.575000,4.130000)(23.300000,20.810000)(23.300000,20.710000)
(23.300000,20.610000)(9.875000,7.230000)(23.225000,5.680000) 3 -3$S_{t_{1}}$3
-33 -3$o$3 -33 -3$\gamma$3 -33 -3$\gamma(t_{1})$3 -3$d_{\max}$3
-3$rd_{\max}$(16.700000,2.150000)
(30.850000,2.250000)(48.200000,8.500000)(33.600000,8.450000)
(19.000000,8.400000)(2.550000,2.050000)(16.700000,2.150000)

(c) At $t_{1}$, $S_{t_{1}}$ has already crossed through $\gamma$, when
$\gamma(t_{1})$ is farthest from $o$. Recall that $r\geq 1$. In turn,
$\gamma^{\prime}_{t_{2}}$ is homotopic in $X$ to $c$.

Figure 2. As $t$ increases from $t_{1}$ to $t_{2}$, and as $\gamma(t)$ goes
farther away from $o$, $S_{T}$ intersects $\gamma$ for some $T\in(0,1)$.

##  2\. Setup

###  2.1. Translating $S_{t}$ to $S^{n-2}_{\operatorname{can}}$

We show here that $S_{t}$ can always be mapped onto
$S^{n-2}_{\operatorname{can}}$ by a scaled isometry $I_{t}$, where

$S^{n-2}_{\operatorname{can}}=\left\\{(a_{1},\dots,a_{n})\in\mathbb{R}^{n}\
\middle|\ \sum_{i=0}^{n-1}a_{i}^{2}=1,a_{n}=0\right\\}.$

The precise choice of $I_{t}$ is not important; we need only that
$I_{\bullet}$ is a continuous map from $S^{1}$ to
$\operatorname{SO}(n)\times\mathbb{R}_{>0}$ such that
$I_{t}(S_{t})=S_{\operatorname{can}}^{n-2}$ for all $t$, but we provide a
concrete description of one such family $I_{\bullet}$ nonetheless. We
construct $I_{t}$ by composing a translation function
$T_{t}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, a rotation function
$R_{t}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ (whose center of rotation is
$o$), and a scaling function $s_{t}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$.
Here is how we construct each of the three functions.

  1. (1)

$\mathbf{T_{t}}$: Let $o^{\prime}$ be the centre of $S_{t}$. We set $T_{t}$ as
the translation of $\mathbb{R}^{n}$ that maps $o^{\prime}$ to $o$.

  2. (2)

$\mathbf{R_{t}}$: Let $\Pi_{t}$ be the hyperplane that contains $S_{t}$, let
$\mathbb{R}^{n}_{0}\colonequals\\{(a_{1},a_{2},\dots,a_{n})\in\mathbb{R}^{n}\mid
a_{n}=0\\}$, and for any $(n-1)$-plane $P\in\mathbb{R}^{n}$, let
$N(P)=(\theta_{1},\theta_{2},\dots,\theta_{n-1},1)$ denote the unit normal
vector of that plane expressed by its $(n-1)$ spherical coordinates.
Additionally, let us assume that $N(\mathbb{R}^{n}_{0})=(0,0,\dots,0,1)$,
without a loss of generality, and let
$N(\Pi_{t})=(\phi_{1},\phi_{2},\dots,\phi_{n-1},1)$. We then set
$R_{t}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ to be the rotation of
$\mathbb{R}^{n}$ about $o$ defined by the equation

$R_{t}((\theta_{1},\theta_{2},\dots,\theta_{n-1},r))=(\theta_{1}-\phi_{1},\theta_{2}-\phi_{2},\dots,\theta_{n-1}-\phi_{n-1},r).$

By this construction, $N\circ R_{t}\circ
T_{t}(\Pi_{t})=N(\mathbb{R}^{n}_{0})$. Thus, since every $(n-1)$-plane passing
through $o$ is uniquely identified by its unit normal vector, it follows that
$R_{t}\circ T_{t}(S_{t})=\mathbb{R}^{n}_{0}$ lies completely in
$\mathbb{R}^{n}_{0}$, and is centred at $o$. Thus, $R_{t}\circ T_{t}(S_{t})$
is a nonzero scaling of $S^{n-2}_{\operatorname{can}}$.

  3. (3)

$\mathbf{s_{t}}$: We set $s_{t}$ as the scaling of $\mathbb{R}^{n}$ for which
$s_{t}\circ(R_{t}\circ
T_{t})(S_{t})=I_{t}(S_{t})=S^{n-2}_{\operatorname{can}}$.

Figures 3, 4(a), and 4(b) illustrate these transformations.

1.000000 -1.000000 (11.366667,3.716667)
(14.766667,3.716667)(10.916667,9.466667)(7.866667,9.466667)
(4.816667,9.466667)(7.966667,3.716667)(11.366667,3.716667)
[fillstyle=solid,fillcolor=diafillcolor,linecolor=diafillcolor]
(11.366667,3.716667)
(14.766667,3.716667)(10.916667,9.466667)(7.866667,9.466667)
(4.816667,9.466667)(7.966667,3.716667)(11.366667,3.716667) 2 -2$o$2 -2$S_{t}$2
-2$o^{\prime}$2 -2$T_{t}$(17.609702,9.176667)
(21.009702,9.176667)(17.159702,14.926667)(14.109702,14.926667)
(11.059702,14.926667)(14.209702,9.176667)(17.609702,9.176667) 2
-2$T_{t}(S_{t})$

Figure 3. The translation $T_{t}$ maps $o^{\prime}$ to $o$.

1.000000 -1.000000 2 -2$o$(17.650000,8.900000)
(21.050000,8.900000)(17.252738,14.706667)(14.202738,14.706667)
(11.152738,14.706667)(14.250000,8.900000)(17.650000,8.900000) 2 -2$R_{t}$2
-2$R_{t}\circ T_{t}(S_{t})$2 -2$T_{t}(S_{t})$

(a) The rotation $R_{t}$ maps $\Pi_{t}$ to $\mathbb{R}^{n}_{0}$.

1.000000 -1.000000 2 -2$o$2 -2$s_{t}\circ R_{t}\circ
T_{t}(S_{t})=S^{n-2}_{can}$2 -22 -2$s_{t}$2 -2$R_{t}\circ T_{t}(S_{t})$

(b) The scaling function $s_{t}$ maps $R_{t}\circ T_{t}(S_{t})$ to
$S^{n-2}_{\operatorname{can}}$.

Figure 4.

Note that whilst $I_{t}$ always maps $S_{t}$ to the same set
$S^{n-2}_{\operatorname{can}}$, it also maps the loop $\gamma$ to a different
loop $\gamma^{\prime}_{t}:=I_{t}(\gamma)$ for each $t\in(0,1)$.

###  2.2. A homotopy lemma

######  Lemma 2.1.

Suppose there is no $t\in(0,1)$ such that $\gamma\cap S_{t}\neq\varnothing$.
Then for any $t,t^{\prime}\in(0,1)$, we have
$\gamma^{\prime}_{t}\simeq\gamma^{\prime}_{t^{\prime}}$ in $X$.

###### Proof.

Let
$F_{t,t^{\prime}}(\cdot,T)\colonequals\gamma^{\prime}_{(1-T)t+(T)t^{\prime}}$.
This is a homotopy taking $\gamma^{\prime}_{t}$ (at $T=0$) to
$\gamma^{\prime}_{t^{\prime}}$ (at $T=1$). For any $T\in[0,1]$,
$F_{t,t^{\prime}}(\cdot,T)$ is simply $\gamma^{\prime}_{t^{\prime\prime}}$ for
some $t^{\prime\prime}\in(0,1)$. Thus, since no curve
$\gamma^{\prime}_{t^{\prime\prime}}$ intersects $X$, $F_{t,t^{\prime}}$ is a
homotopy in $X$. ∎

##  3\. Finding $t_{1}$

######  Notation 3.1.

Throughout the rest of the paper, we let $c$ denote the constant path at $o$.

Let us choose $t_{1}$ such that $\|\gamma(t_{1})-o\|:=d_{\max}$ is maximal. We
shall prove the following lemma regarding $t_{1}$:

######  Lemma 3.2.

Assume there is no $t\in(0,1)$ such that $S_{t}$ intersects $J$. Then
$\gamma^{\prime}_{t_{1}}$ is homotopic in $X$ to $c$.

The idea is as follows. Let $x$ denote an arbitrary point in $S_{t_{1}}$. Then
we have $\|x-o\|=rd_{\max}$, since $r\geq 1$. Then
$\|\gamma(t)-o\|\leq\|x-o\|$ for any $t\in(0,1)$. Thus, $S_{t_{1}}$ has
already slipped out from $\gamma$, being far enough from $\gamma$ that
$\gamma$ can be shrunk to $c$ by a linear homotopy $L$ that does not intersect
$S_{t}$. See Figure 5.

1.000000 -1.000000 (23.175000,5.980000)
(36.525000,4.430000)(23.250000,21.110000)(23.250000,21.010000)
(23.250000,20.910000)(9.825000,7.530000)(23.175000,5.980000)
[fillstyle=solid,fillcolor=diafillcolor,linecolor=diafillcolor]
(23.175000,5.980000)
(36.525000,4.430000)(23.250000,21.110000)(23.250000,21.010000)
(23.250000,20.910000)(9.825000,7.530000)(23.175000,5.980000) 2 -2$S_{t_{1}}$2
-22 -2$o$2 -22 -2$\gamma$2 -22 -2$\gamma(t_{1})$2 -2$d_{max}$2 -2$rd_{max}$

(a) As $r\geq 1$, $rd_{\max}\geq d_{\max}$. Thus, $S_{t_{1}}$ is so far that
it cannot intersect $\gamma$ as it is homotoped to $c$ at $o$.

1.000000 -1.000000 (22.450000,5.850000)
(35.800000,4.300000)(22.525000,20.980000)(22.525000,20.880000)
(22.525000,20.780000)(9.100000,7.400000)(22.450000,5.850000)
[fillstyle=solid,fillcolor=diafillcolor,linecolor=diafillcolor]
(22.450000,5.850000)
(35.800000,4.300000)(22.525000,20.980000)(22.525000,20.880000)
(22.525000,20.780000)(9.100000,7.400000)(22.450000,5.850000) 2
-2$S_{t_{1}}$(22.550000,12.250000)
(27.250000,11.450000)(22.477009,20.802871)(22.477009,20.702871)
(22.477009,20.602871)(17.850000,13.050000)(22.550000,12.250000)
(22.600000,9.050000)
(31.300000,7.750000)(22.443289,20.941421)(22.443289,20.841421)
(22.443289,20.741421)(13.900000,10.350000)(22.600000,9.050000) 2 -22 -2$o$2
-2$t=0$2 -22 -2$t=\frac{1}{4}$2 -2$t=\frac{1}{2}$1 -1$\gamma$

(b) $\gamma$ being homotoped to $c$ by a straight line homotopy.

Figure 5.

###### Proof of Lemma 3.2.

We first show that the straight-line homotopy
$L:[0,1]\times[0,1]\rightarrow\mathbb{R}^{n}$ from $\gamma$ to $c$ never
intersects $S_{t_{1}}$. We consider two cases for $L(s,\cdot)$; first, when
$t=0$, and second, when $t\in(0,1]$.

  * •

$t=0$: $L(s,0)=\gamma$ does not intersect $S_{t_{1}}$ for any $s\in[0,1]$
because of our assumption that no such intersection occurs.

  * •

$t\in(0,1]$: Since $L$ is a straight-line homotopy to $o$, $\|L(s,\cdot)-o\|$
must be monotonically decreasing over $(0,1]$, so in particular

$\|L(s,t)-o\|<d_{\max}\leq\|x-o\|.$

That is, the distance of $L(s,t)$ from $o$ is always less than that of $x$
from $o$. Thus, $L$ never intersects $S_{t_{1}}$ when $t\in(0,1]$.

We conclude that $L(s,t)$ does not intersect $S_{t_{1}}$ for any
$s,t\in[0,1]$.

Thus $I_{t_{1}}$ induces the desired homotopy $I_{t_{1}}\circ L$ taking
$I_{t_{1}}\circ\gamma=\gamma^{\prime}_{t_{1}}$ to $I_{t_{1}}\circ
c\colonequals c^{\prime}$. Since $L$ never intersects $S_{t_{1}}$, and since
$I_{t_{1}}$ is a bijection, $I_{t_{1}}\circ L$ never intersects
$I_{t_{1}}(S_{t_{1}})=S^{n-2}_{\operatorname{can}}$. Therefore,
$\gamma^{\prime}_{t_{1}}\simeq c^{\prime}$ in $X$ via the homotopy
$I_{t_{1}}\circ L$. ∎

##  4\. Finding $t_{2}$

######  Lemma 4.1.

Let $\theta_{v}$ be the angle of the vertex of $v$ the triangle $\triangle$ to
be inscribed. Suppose there is no $t\in(0,1)$ such that $\gamma$ intersects
$S_{t}$. If $\theta_{v}$ is such that

$\limsup_{\delta\rightarrow
0^{+}}\Theta_{\delta}<\theta_{v}<\liminf_{\delta\rightarrow
0^{+}}\Theta^{\prime}_{\delta},$

then there exists some $t_{2}\in(0,1)$ has the property that
$\gamma^{\prime}_{t_{2}}\not\simeq c^{\prime}$ in $X$.

The proof involves proving a series of lemmata (namely, Lemmata 4.2 to 4.15)
leading to a proof of Lemma 4.1. We begin by recalling a well-known and simple
preliminary result.

######  Lemma 4.2.

Let $A,B\subset\mathbb{R}^{n}$ be two disjoint compact sets. There is some
$\inf(A,B)>0$ such that $\|a-b\|\geq\inf(A,B)$ for any $a\in A$ and any $b\in
B$.

For each of the remaining lemmata (from Lemma 4.3 to Lemma 4.15), we assume
the same hypotheses as are assumed for Lemma 4.1, namely that there is no
$t\in(0,1)$ such that $\gamma$ intersects $S_{t}$, and that the angle
$\theta_{v}$ of some vertex $v$ of $\triangle$ is such that
$\limsup_{\delta\rightarrow
0^{+}}\Theta_{\delta}<\theta_{v}<\liminf_{\delta\rightarrow
0^{+}}\Theta^{\prime}_{\delta}$.

######  Lemma 4.3.

There exists some $\varepsilon>0$ such that
$\sup\Theta_{\varepsilon}<\theta_{v}<\inf\Theta^{\prime}_{\varepsilon}$.

###### Proof.

Since $\limsup_{\delta\rightarrow
0^{+}}\Theta_{\delta}<\theta_{v}<\liminf_{\delta\rightarrow
0^{+}}\Theta^{\prime}_{\delta}$, there exist arbitrarily small values of
$\varepsilon_{1},\varepsilon_{2}>0$ such that
$\sup\Theta_{\varepsilon_{1}}<\theta_{v}<\inf\Theta^{\prime}_{\varepsilon_{2}}$.
Let $\varepsilon\colonequals\min(\varepsilon_{1},\varepsilon_{2})$. Then,
$\sup\Theta_{\varepsilon}<\theta_{v}<\inf\Theta^{\prime}_{\varepsilon}$. ∎

######  Lemma 4.4.

For a positive real number $r$, let $B(o,r)$ be the closed $n$-ball of radius
$r$, centred at $o$. Additionally, let the neighbourhood
$N\colonequals\operatorname{Im}\gamma\mid_{(1-\varepsilon,\varepsilon)}$. Then
there exists $d_{\min}>0$ such that $J\cap B(o,d_{\min})\subset N$.

Note that $(1-\varepsilon,\varepsilon)$ is taken modulo 1, i.e.
$(1-\varepsilon,\varepsilon)=[0,\varepsilon)\cup(1-\varepsilon,1]$.

###### Proof.

Let $d_{\min}=\frac{\inf(\\{o\\},J-N)}{2}$. Since $\\{o\\}$ and $J-N$ are
disjoint and compact, Lemma 4.2 tells us that $d_{\min}>0$. Since
$d_{\min}<\inf(\\{o\\},J-N)$, it thus follows that $(J-N)\cap
B(o,d_{\min})=\varnothing$, and thus that $J\cap B(o,d_{\min})\subset N$. ∎

######  Lemma 4.5.

There exists some $t_{2}\in[0,1]$ such that $S_{t_{2}}\subset\partial
B(o,d_{\min})$.

###### Proof.

Let $p\in N$ be a point such that $\|p-o\|=d\colonequals\frac{d_{\min}}{r}$.
We let $t_{2}\in(0,1)$ be such that $p=\gamma(t_{2})$. Then all the points of
$S_{t_{2}}$ lie at a distance $rd=d_{\min}$ from $o$. Therefore
$S_{t_{2}}\subset\partial B(o,d_{\min})$. ∎

######  Definition 4.6.

We say a point lies in _the interior (or exterior) of $S_{t_{2}}$_ when it
lies within _the interior (or exterior) of $S_{t_{2}}$ when $\Pi_{t_{2}}$ is
taken as its ambient space_.

######  Lemma 4.7.

The interior of $S_{t_{2}}$ lies in $B(o,d_{\min})$.

###### Proof.

Since (from Lemma 4.5) $S_{t_{2}}\subset\partial B(o,d_{\min})$, it follows
that its interior is a subset of $B(o,d_{\min})$ too. ∎

######  Lemma 4.8.

All intersections of $J-N$ with $\Pi_{t_{2}}$ lie in the exterior of
$S_{t_{2}}$.

###### Proof.

Assume that some $p\in(J-N)\cap\Pi_{t_{2}}$ lies in the interior of
$S_{t_{2}}$. Since (from Lemma 4.7) the interior of $S_{t_{2}}$ is a subset of
$B(o,d_{\min})$, it follows that $p\in(J-N)\cap B(o,d_{\min})$. However, from
Lemma 4.4, $J\cap B(o,d_{\min})\subset N$, contradicting the assumption that
$p$ lies in the interior of $S_{t_{2}}$. ∎

See Figure 6 for an illustration of the preceding Lemmata.

1.000000 -1.000000 (18.308300,7.650000)
(22.108300,10.550100)(25.783012,10.575000)(18.583012,12.775000)
(18.483012,12.825000)
(12.683012,15.491700)(27.116700,13.166700)(18.900100,17.216700) 2
-2$o$(19.375000,3.583330)
(15.975000,3.850000)(8.958430,5.466670)(16.908400,-0.283333) 2
-2$p^{\prime}$(-0.750000,16.700000)
(3.633012,19.250000)(35.933012,5.500000)(31.450000,3.350000)
(26.966988,1.200000)(-5.133012,14.150000)(-0.750000,16.700000)
(18.286900,7.572210)
(16.736900,5.922210)(28.525100,2.933330)(19.475000,3.633330) 2 -2$\theta$2
-2$S_{t_{2}}$2 -2$N$2 -2$p$2 -2$d_{min}$(18.900100,17.200000)
(17.350100,18.350000)(22.916700,17.316700)(19.700100,18.700000)
(19.850100,18.600000)
(18.250100,19.900000)(19.900100,18.850000)(20.029800,21.220200) 2
-2$B(o,d_{min})$2 -2$b$2 -2$J-N$(19.800000,21.775000)
(18.108400,23.416700)(22.358400,21.666700)(19.658400,23.466700)
(19.870400,23.226600)
(15.516700,27.666700)(30.116700,25.266700)(21.216700,28.266700)
(21.188000,28.223000)
(-3.283330,34.016700)(43.616700,27.566700)(21.846500,33.589600)
(21.912100,33.576700)
(-17.108300,41.966700)(56.458400,26.266700)(23.016700,43.266700) 2 -2$J-N$2
-2$b^{\prime}$2 -2$\inf\Theta^{\prime}_{\varepsilon}$

Figure 6. Here, the angle between the blue lines is
$\sup\Theta_{\varepsilon}$. The angle between the left-facing blue line and
the left-facing black line is $\inf\Theta^{\prime}_{\varepsilon}$. Since
$\sup\Theta_{\varepsilon}<\theta_{v}<\inf\Theta^{\prime}_{\varepsilon}$, it
follows $S_{t_{2}}$ is wide enough such that $N$ always intersects
$\Pi_{t_{2}}$ at the interior of $S_{t_{2}}$. Here, $p^{\prime}$ denotes one
such point of intersection.

Since $I_{t_{2}}$ is an isometry, all of the Lemmata above (which apply for
$\gamma$) also apply to the curve
$\gamma^{\prime}_{t_{2}}=I_{t_{2}}\circ\gamma$—for example,
$\gamma^{\prime}_{t_{2}}\mid_{[1-\varepsilon,\varepsilon]}$ intersects
$I_{t_{2}}(\Pi_{t_{2}})=\mathbb{R}^{n}_{0}$ only at the exterior of
$I_{t_{2}}(S_{t_{2}})=S^{n-2}_{\operatorname{can}}$.

######  Definition 4.9.

Let $P:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}\times\mathbb{R}$ be
defined by

$P((r_{1},r_{2},\dots,r_{n}))=(d(r),r_{n}),$

where $d:\mathbb{R}^{n-1}\rightarrow\mathbb{R}_{\geq 0}$ is defined to be

$d(r)=\sqrt{r_{1}^{2}+r_{2}^{2}+\cdots+r_{n-1}^{2}}.$

######  Definition 4.10.

For any given $p\in\mathbb{R}^{2}\setminus\operatorname{Im}f$, let a path
$f:[0,1]$ be parametrized to polar form $(r(t),\theta(t))$, where
$r(t)=\|f(t)-p\|$ and $\theta(t)$ continuously maps $t$ to the angle of the
segment $pf(t)$ relative to the positive vertical axis from $p$. Then, we
define the winding number $\eta_{p}(f)$ of $f$ relative to $p$ by the equation

$\frac{\theta(1)-\theta(0)}{2\pi}.$

######  Lemma 4.11.

$\eta_{(1,0)}(P^{\prime}\circ\gamma^{\prime}_{t_{2}}\mid_{[1-\varepsilon,\varepsilon]})<0$.

###### Proof.

To begin with, note that $P(S^{n-2}_{\operatorname{can}})=(1,0)$. Let
$(0,1)=u$, and let $\phi(t)$ be the angle that the segment
$uP\circ\gamma^{\prime}_{t_{2}}(t)$ makes relative to the segment
$uP\circ\gamma^{\prime}_{t_{2}}(0)$. See Figure 7.

Assume, for the sake of contradiction, that
$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{[1-\varepsilon,0]})>0$ Then,
since $\sup\Theta_{\varepsilon}<\theta_{v}$, it follows
$\phi(\varepsilon)<\theta_{v}$. Also, since
$\inf\Theta^{\prime}_{\varepsilon}>\theta_{v}$, it follows that
$\phi(1-\varepsilon)>\theta_{v}$.

1.000000 -1.000000 2 -2$P\circ\gamma^{\prime}_{t_{2}}(0)$2 -22 -2$(1,0)$2 -22
-2$\sup\Theta_{\varepsilon}$2 -2$\inf\Theta^{\prime}_{\varepsilon}$2
-2$\gamma^{\prime}_{t_{2}}(\varepsilon)$2
-2$\gamma^{\prime}_{t_{2}}(1-\varepsilon)$(15.845190,5.269454)
(15.450000,6.350000)(17.394454,5.619454)(15.425000,13.000000)
(15.619454,13.469454)
(15.500000,16.250000)(22.600000,15.000000)(24.870190,12.344454) 2
-2$P\circ\gamma^{\prime}_{t_{2}}\mid_{[\varepsilon,1-\varepsilon]}$

Figure 7. The path
$P\circ\gamma^{\prime}_{t_{2}}\mid_{[1-\varepsilon,\varepsilon]}$. This path
cannot have a negative winding number, since that would contradict the fact
that $\phi(t)\geq\inf\Theta^{\prime}_{\varepsilon}$ for all
$t\in[1-\varepsilon,0]$ and $\phi(t)\leq\sup\Theta_{\varepsilon}$ for all
$t\in[0,\varepsilon]$.

As $\phi(1-\varepsilon)>\phi(\varepsilon)$, if
$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{[1-\varepsilon,1-\varepsilon]})>0$
(i.e. if the net movement is anticlockwise around $p$) then
$\phi(t)=\theta_{v}$ for some $t\in[\varepsilon,1-\varepsilon]$, in
contradiction to the fact that $\phi(t)\geq\inf\Theta^{\prime}_{\varepsilon}$
for all $t\in[1-\varepsilon,0]$ $\phi(t)\leq\sup\Theta_{\varepsilon}$ for all
$t\in[0,\varepsilon]$. ∎

######  Definition 4.12.

Let $a_{n}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be the last-coordinate map,
i.e. $a_{n}((r_{1},r_{2},\ldots,r_{n}))=r_{n}$.

######  Lemma 4.13.

$a_{2}\circ P\circ\gamma^{\prime}_{t_{2}}(\varepsilon)>0$.

###### Proof.

Assume that this is not true. Then, it would follow that
$\|o-\gamma^{\prime}_{t_{2}}(\varepsilon)\|<\|o-u\|$, which implies that
$\gamma^{\prime}_{t_{2}}(\varepsilon)$ lies in the interior of
$B(o,d_{\min})$. This would, in turn, also imply that $J_{N}$ (in whose
closure $\gamma^{\prime}_{t_{2}}(\varepsilon)$ belongs) has a nonempty
intersection with $B(o,d_{\min})$. However, this contradicts the definition of
$N$ in Lemma 4.4. ∎

######  Lemma 4.14.

$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{[\varepsilon,1-\varepsilon]})<0$.

###### Proof.

Let $p_{0}=(a,0)$, where $0\leq a<1$. Then, the angle that $up_{0}$ makes
against the positive vertical axis from $u$ is $\frac{\pi}{2}$. From Lemma
4.13, we infer that $\phi(\varepsilon)<\frac{\pi}{2}$, and thus that
$\phi(\varepsilon)-\frac{\pi}{2}<0$. See Figure 8.

1.000000 -1.000000 2 -22 -2$(1,0)$2 -22
-2$P\circ\gamma^{\prime}_{t_{2}}(\varepsilon)$2
-2$P\circ\gamma^{\prime}_{t_{2}}(1-\varepsilon)$(16.400000,5.075000)
(16.004800,6.155550)(22.200000,8.325000)(25.100000,11.875000)
(15.850000,5.587500)
(14.850000,14.475000)(22.600000,15.000000)(24.870200,12.344500)
(16.400000,5.075000)
(16.004800,6.155550)(26.400000,5.425000)(25.100000,11.875000)
(15.845190,5.269454)
(15.449990,6.350004)(15.705868,21.758368)(24.800000,12.575000)
(15.845190,5.269454)
(14.850000,11.825000)(16.118799,22.054982)(24.868799,12.404982)
(16.400000,5.075000)
(20.700000,4.875000)(28.250000,5.775000)(25.100000,11.875000) 2
-2$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{\varepsilon,1-\varepsilon})<0$2
-2$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{\varepsilon,1-\varepsilon})>0$

Figure 8.
$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{\varepsilon,1-\varepsilon})>0$
if and only if the curve passes $(0,1)$ by the left, whilst
$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{\varepsilon,1-\varepsilon})<0$
if and only if it passes it from the right.

Since we also know that $\frac{\pi}{2}-\phi(1-\varepsilon)<0$, it follows that
if
$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{[1-\varepsilon,\varepsilon]})>0$,
then there is some $t^{\prime}\in[\varepsilon,1-\varepsilon]$ such that
$\phi(t^{\prime})=\frac{\pi}{2}$, i.e. that
$P\circ\gamma^{\prime}_{t_{2}}(t^{\prime})=(a,0)$ for some $0\leq a<1$.
However, it would then follow that
$\gamma^{\prime}_{t_{2}}(t^{\prime})\in\gamma^{\prime}_{t_{2}}\mid_{[\varepsilon,1-\varepsilon]}=J-N$
lies within the interior of $S^{n-2}_{can}$, which is in contradiction to
Lemma 4.8. ∎

######  Lemma 4.15.

$P\circ\gamma^{\prime}_{t_{2}}\not\simeq P\circ c^{\prime}\colonequals
c^{\prime\prime}$ in $P(X)=\mathbb{R}^{2}-(1,0)$.

###### Proof.

Since
$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}})=\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{[1-\varepsilon,\varepsilon]})+\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}}\mid_{[\varepsilon,1-\varepsilon]})$,
Lemmata 4.11 and 4.14 imply that
$\eta_{(1,0)}(P\circ\gamma^{\prime}_{t_{2}})\neq 0$, which is true if and only
if this curve is not nullhomotopic. ∎

###### Proof of Lemma 4.1.

Assume that $\gamma^{\prime}_{t_{2}}\simeq c^{\prime}$. Let
$H:[0,1]\times[0,1]\rightarrow\mathbb{R}_{\geq 0}\times\mathbb{R}$ be a
homotopy such that $H(0,\cdot)=\gamma^{\prime}_{t_{2}}$ and
$H(1,\cdot)=c^{\prime}$. Then, the homotopy $P\circ H$ takes $P\circ
H(0,\cdot)=P\circ\gamma^{\prime}_{t_{2}}$ to the constant path $P\circ
H(1,\cdot)=c^{\prime\prime}$. However, from Lemma 4.15, we know that this is
impossible. ∎

Therefore, Lemmata 3.2 and 4.1 immediately imply Theorem 1.3.

As a corollary of independent interest of this theorem, we get the following.

######  Theorem 4.16.

If $o\in J$ is a differentiable point, then any triangle $\triangle$ can be
inscribed in $J$.

###### Proof.

When $o$ is differentiable, $\limsup_{\delta\rightarrow
0^{+}}\Theta_{\delta}=0$ and $\liminf_{\delta\rightarrow
0^{+}}\Theta^{\prime}_{\delta}=\pi$. Then, the angle $\theta_{v}$ of any
vertex of $\triangle$ always satisfies $\limsup_{\delta\rightarrow
0^{+}}\Theta_{\delta}<\theta_{v}<\liminf_{\delta\rightarrow
0^{+}}\Theta^{\prime}_{\delta}$. Thus, $\triangle$ always satisfies the
requirements of Theorem 1.3 to be inscribed in $J$. ∎

##  5\. Generalising theorem 1.2

We now prove that subject to certain restrictions, Theorem 1.2 can be
generalised to any given Jordan curve $J\subset\mathbb{R}^{n}$. We state this
condition before stating our generalisation.

######  Definition 5.1.

For any given $o\in J$, and any $p\in J-o$, let
$a_{p}:[0,1]\rightarrow\mathbb{R}$ be defined as

$a_{p}(t)=\frac{\vv{o\gamma(t)}}{\|o\gamma(t)\|}\cdot\vv{op}.$

If there is some $\varepsilon>0$ such that
$a_{p}\mid_{(1-\varepsilon,\varepsilon)}$ is monotone for each
$p\in\operatorname{Im}\gamma\mid_{(1-\varepsilon,\varepsilon)}-o$, we say that
$J$ is _strongly monotone_ at $o$. Here,
$\gamma\mid_{(1-\varepsilon,\varepsilon)}$ is called a _strongly locally
monotone neighbourhood_ of $o$.

Examples wherein similar conditions have been used to prove inscription
theorems can be found in, for example, [Str89]. See Figure 9 for examples of
neighbourhoods that both satisfy and fall outside this condition.

1.000000 -1.000000 2 -2$\gamma(1-\varepsilon)$2 -2$\gamma(\varepsilon)$2
-2$o$2 -2$p$

(a) $a_{p}\mid_{(1-\varepsilon,\varepsilon)}$ is monotone for any
$p\in\gamma\mid_{(1-\varepsilon,\varepsilon)}-o$.

1.000000 -1.000000 2 -2$\gamma(1-\varepsilon)$2 -2$\gamma(\varepsilon)$2
-2$o$2 -2$p$

(b) $a_{p}\mid_{(1-\varepsilon,0]}$ is monotone decreasing but
$a_{p}\mid_{[0,\varepsilon)}$ is monotone increasing, so
$a_{p}\mid_{(1-\varepsilon,\varepsilon)}$ can’t be monotone for any
$p\in\gamma\mid_{(1-\varepsilon,\varepsilon)}-o$.

Figure 9. $J$ is strongly locally monotone at $o$ in Figure 9(a), and is not
in Figure 9(b).

######  Theorem 5.2.

If $J$ is strongly locally monotone at $o$, then the point $o\in J$ has an
inscribed inscribed equilateral triangle.

Note that, in comparison to Theorem 1.3 for an equilateral triangle, this
theorem is stronger, since it allows for an unaccountably infinite number of
equilateral triangles to be inscribed at the strongly monotone neighbourhood
containing the point $o$, since any point of that neighbourhood can be chosen
as the first point for the equilateral triangle. By contrast, finding a
suitable neighbourhood that satisfies the hypothesis of Theorem 1.3 only
ensures at least a single triangle of the desired specifications can be
inscribed in it.

For the proof, our first concern is how we identify when $o$ inscribes an
equilateral triangle. Let $\phi_{s}:[0,1]\rightarrow[0,s]$ be defined by
$\phi_{s}(t)=st$. For all $s\in(0,1)$, let
$r_{1,s}:[0,1]\rightarrow\mathbb{R}_{\geq 0}$ and
$r_{2,s}:[0,1]\rightarrow\mathbb{R}_{\geq 0}$ be respectively defined by

$r_{1,s}(t)=\frac{\|\gamma\circ\phi_{s}(t)-o\|}{\|\gamma(s)-o\|}\text{ and
}r_{2,s}(t)=\frac{\|\gamma\circ\phi_{s}(t)-\gamma(s)\|}{\|\gamma(s)-o\|},$

where, for any $t\in[0,1]$, $r_{1,s}(t)$ and $r_{2,s}(t)$ respectively
represent the ratio of the lengths of the sides $o\gamma\circ\phi_{s}(t)$ and
$\gamma(s)\gamma\circ\phi_{s}(t)$ to the side $o\gamma(s)$. Then, the _ratio
path_ $R_{s}:[0,1]\rightarrow\mathbb{R}^{2}$ for any $s\in(0,1)$ is

$R_{s}(t)=(r_{1,s}(t)-1,r_{2,s}(t)-1).$

To prove Theorem 5.2, then, it suffices to show that, for some
$s^{\prime},t^{\prime}\in(0,1)$,
$r_{1,s^{\prime}}(t^{\prime})=r_{2,s^{\prime}}(t^{\prime})=1$, and thus that
$R_{s^{\prime}}(t^{\prime})=(0,0)$. Note that $R_{s}(0)=(-1,0)$ and
$R_{s}(1)=(0,-1)$.

######  Definition 5.3.

For any pair of paths $f,g$, let $f\ast g$ denote the concatenation of the
paths $f,g$.

######  Definition 5.4.

For any path $f$, let $\bar{f}$ denote the inverse of $f$, defined by
$\bar{f}(t)=f(1-t)$.

######  Definition 5.5.

For any $s,s^{\prime}\in(0,1)$, let the loop $L_{s,s^{\prime}}$ be defined by
the equation $L_{s,s^{\prime}}\colonequals R_{s}\ast\bar{R}_{s^{\prime}}$.

The approach is as follows. Let $c^{\prime\prime}$ denote the constant path at
$R_{s}(0)=(-1,0)$. To begin, we use Lemma 5.6 to prove Lemma 5.7, which states
that the loop $L_{s,s^{\prime}}\simeq c^{\prime\prime}$ in
$\mathbb{R}^{2}-(0,0)$ for any $s,s^{\prime}\in(0,1)$ if $R_{s}$ doesn’t
contain $(0,1)$ for any $s\in(0,1)$. Then, we combine two technical
lemmata—namely, Lemmata 5.8 and 5.9—to prove Lemma 5.10, which states that
$L_{s_{1},s_{2}}\not\simeq c^{\prime\prime}$ if $R_{s}$ doesn’t contain
$(0,0)$ for any $s\in(0,1)$. However, from Lemma 5.7, we also know that
$L_{s_{1},s_{2}}\simeq c^{\prime\prime}$ if $R_{s}$ doesn’t contain $(0,0)$
for any $s\in(0,1)$. Since both of these results follow from the same
hypothesis—namely, that $R_{s}$ doesn’t contain $(0,0)$ for any $s\in(0,1)$—we
know then that this hypothesis is false. A proof of Theorem 5.2 then
immediately follows.

######  Lemma 5.6.

Assume $R_{s}(t)$ does not contain $(0,0)$ for any $s\in(0,1)$. Then,
$R_{s}\simeq R_{s^{\prime}}$ in $\mathbb{R}^{2}-(0,0)\colonequals X^{\prime}$
for any $s,s^{\prime}\in(0,1)$.

###### Proof.

Let $F^{\prime}_{s,s^{\prime}}(\cdot,T)\colonequals R_{s(1-T)+s^{\prime}T}$.
This is a homotopy taking $R_{s}$ (at $T=0$) to $R_{s^{\prime}}$ ($T=1$).
Then, for each $T\in[0,1]$, $F^{\prime}_{s,s^{\prime}}(\cdot,T)$ is simply
$R_{s^{\prime\prime}}$ for some $s^{\prime\prime}\in[s,s^{\prime}]$. Thus, if
no curve $R_{s^{\prime\prime}}$ intersects $(0,0)$, then $F_{s,s^{\prime}}$ is
the desired homotopy taking $R_{s}$ to $R_{s^{\prime}}$ in $X^{\prime}$. ∎

######  Lemma 5.7.

Assume $R_{s}$ not contain $(0,0)$ for any $s\in(0,1)$. Then
$L_{s,s^{\prime}}\simeq c^{\prime\prime}$ in $X^{\prime}$ for any
$s,s^{\prime}\in(0,1)$.

###### Proof.

From Lemma 5.6, $R_{s}\simeq R_{s^{\prime}}$ if $R_{s}$ does not contain
$(0,0)$ for any $s\in(0,1)$. Then,
$L_{s,s^{\prime}}=R_{s}*\bar{R}_{s^{\prime}}\simeq
R_{s^{\prime}}*\bar{R}_{s^{\prime}}\simeq c^{\prime\prime}$. ∎

######  Lemma 5.8.

There exists some $s_{1}\in(0,1)$ such that $R_{s_{1}}\subset\mathbb{R}_{\leq
0}\times\mathbb{R}$.

###### Proof.

Let $s_{1}\in(0,1)$ be such that $\|\gamma(s_{1})-o\|$ is maximal. Then, by
definition, $\|\gamma(t)-o\|\leq\|\gamma(s_{1})-o\|$ for all $t\in[0,1]$, and
thus
$r_{1,s_{1}}(t)=\frac{\|\gamma\circ\phi_{s_{1}}(t)-o\|}{\|\gamma(s_{1})-o\|}\leq
1$. Let $a_{1}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ map each point to its
first component. Then, $a_{1}\circ R_{s_{1}}(t)=r_{1,s_{1}}(t)-1\leq 0$, and
thus $R_{s_{1}}\subset\mathbb{R}_{\leq 0}\times\mathbb{R}$. ∎

######  Lemma 5.9.

Let $J$ be strongly locally monotone at $o$. There exists some $s_{2}\in(0,1)$
such that $R_{s_{2}}\cap(\mathbb{R}_{<0}\times\mathbb{R}_{<0})=\varnothing$.

###### Proof.

Let $\varepsilon>0$ be such that
$\gamma\mid_{(1-\varepsilon,\varepsilon)}\colonequals U\subset J$ is a
strongly locally monotone neighbourhood of $o$. Let
$d\colonequals\frac{\inf(\\{o\\},J-U)}{3}$, and let $s_{2}\in(0,\varepsilon)$
be such that $\|\gamma(s_{2})-o\|=d$. Note that
$J-U\subset\gamma\mid_{[0,s_{2}]}$. Also, let $B^{\prime}(o,r)$ denote the
open $n$-ball of radius $r$ centred at $o$.

Then, $R_{s_{2}}(t)\in\mathbb{R}_{<0}\times\mathbb{R}_{<0}$ if and only if
$\|\gamma(t)-o\|<d$ and $\|\gamma(t)-\gamma(s_{2})\|<d$ for all $t\in[0,1]$,
and thus, if and only if $\gamma\circ\phi_{s_{2}}\cap B\neq\varnothing$, where
$B\colonequals B^{\prime}(o,d)\cap B^{\prime}(\gamma(s_{2}),d)$. As such, it
shall suffice to show that $\gamma\circ\phi_{s_{2}}\cap B=\varnothing$ to
complete the proof. We consider three parts of $\gamma\circ\phi_{s_{2}}$
separately:$\gamma\circ\phi_{s_{2}}\mid_{[0,\frac{\varepsilon}{s_{2}}]}$,
$\gamma\circ\phi_{s_{2}}\mid_{[\frac{\varepsilon}{s_{2}},\frac{1-\varepsilon}{s_{2}}]}=J-U$,
and $\gamma\circ\phi_{s_{2}}\mid_{[\frac{1-\varepsilon}{s_{2}},1]}$.

  * •

$\gamma\circ\phi_{s_{2}}\mid_{[0,\frac{\varepsilon}{s_{2}}]}:$ Since (without
a loss of generality) $a_{\gamma(s_{2})}$ is monotonically increasing, the
starting point $o$ of
$\gamma\circ\phi_{s_{2}}\mid_{[0,\frac{\varepsilon}{s_{2}}]}$ is its lowest
point. Thus, since $a_{\gamma(s_{2})}(o)>a_{\gamma(s_{2})}(b)$ for any $b\in
B$, no point of $\gamma\circ\phi_{s_{2}}\mid_{[0,\frac{\varepsilon}{s_{2}}]}$
lies close enough to the axis defined by $\vv{o\gamma(s_{2})}$ to intersect
$B$. See Figure 10 for a diagram that illustrates this.

1.000000 -1.000000 (22.854800,12.495200)
(30.200000,10.450000)(14.250000,6.550000)(26.900000,6.650000)
(22.887120,24.291840)
(14.787120,26.341840)(23.784400,29.300000)(24.050000,29.300000) 2 -2$o$2
-2$\gamma\circ\phi_{s_{2}}(1)=\gamma(s_{2})$2 -2$B$2
-2$\gamma\mid_{[s_{2},0]}$2 -2$B^{\prime}(o,d)$2
-2$B^{\prime}(\gamma(s_{2}),d)$

Figure 10. No point of
$\gamma\circ\phi_{s_{2}}\mid_{[0,\frac{\varepsilon}{s_{2}}]}$ lies low enough
to intersect $B$, and no point of
$\gamma\circ\phi_{s_{2}}\mid_{[\frac{1-\varepsilon}{s_{2}},1]}$ is high enough
to intersect $B$.

  * •

$\gamma\circ\phi_{s_{2}}\mid_{[\frac{\varepsilon}{s_{2}},\frac{1-\varepsilon}{s_{2}}]}=J-U$:
Since $d<\inf(\\{o\\},J-U)$, it follows that $(J-U)\cap
B^{\prime}(o,d)=\varnothing$. Then

$\inf(\\{\gamma(s_{2})\\},J-U)\geq\|\inf(\\{o\\},J-U)\|-\|\gamma(s_{2})-o\|=3d-d=2d.$

Since $d<2d\leq\inf(\gamma(s_{2}),J-U)$, it follows that $(J-U)\cap
B^{\prime}(\gamma(s_{2}),d)=\varnothing$. Therefore, $J-U$ intersects neither
$B^{\prime}(o,d)$ nor $B^{\prime}(\gamma(s_{2}),d)$, and in particular,
$(J-U)\cap B=\varnothing$.

  * •

$\gamma\circ\phi_{s_{2}}\mid_{[\frac{1-\varepsilon}{s_{2}},1]}$: Since
(without a loss of generality) $a_{\gamma(s_{2})}$ is monotonically
increasing, the end point $\gamma\circ\phi_{s_{2}}(1)=\gamma(s_{2})$ of
$\gamma\circ\phi_{s_{2}}\mid_{[\frac{1-\varepsilon}{s_{2}},1]}$ is its
“highest” point. That is to say,
$a_{s_{2}}\circ\gamma(s_{2})>a_{\gamma(s_{2})}\circ\gamma\circ\phi_{s_{2}}\mid_{[\frac{1-\varepsilon}{s_{2}},1]}(t)$
for any $t$. Thus, since $a_{\gamma(s_{2})}(o)<a_{\gamma(s_{2})}(b)$ for any
$b\in B$ and any $t$,
$a_{\gamma_{s_{2}}}\circ\gamma\circ\phi_{s_{2}}\mid_{[0,\frac{\varepsilon}{s_{2}}]}(t)<a_{\gamma(s_{2})}(b)$.
In other words, no point of
$\gamma\circ\phi_{s_{2}}\mid_{[0,\frac{\varepsilon}{s_{2}}]}$ lies high enough
(relative to the axis defined by $\vv{o\gamma(s_{2})}$) to intersect $B$.
Again, see Figure 10.

Therefore, $\gamma\circ\phi_{s_{2}}\cap B=\varnothing$, and thus
$R_{s_{2}}\cap(\mathbb{R}_{<0}\times\mathbb{R}_{<0})=\varnothing$. ∎

######  Lemma 5.10.

If $R_{s}$ doesn’t contain $(0,0)$ for any $s\in(0,1)$, then
$L_{s_{1},s_{2}}\not\simeq c^{\prime\prime}$ in $X$.

###### Proof.

Since neither $R_{s_{1}}$ nor $R_{s_{2}}$ contain the origin,
$L_{s_{1},s_{2}}$ has a well-defined winding number around $(0,0)$. We show
that the winding number $\eta(L_{s_{1},s_{2}})$ of $L_{s_{1},s_{2}}$ around
$(0,0)$ is 1. As $\eta(c^{\prime\prime})=0$, and as the winding number of a
curve is a homotopy invariant, demonstrating this shall be sufficient for this
proof.

For any $t\in[0,1]$, let $\theta(t)$ be equal to the angle of
$L_{s_{1},s_{2}}(t)$ relative to the $x$-axis. To compute
$\eta(L_{s_{1},s_{2}})$, we first find $\theta(\frac{1}{2})-\theta(0)$ and
$\theta(1)-\theta(\frac{1}{2})$.

  * •

$\theta(\frac{1}{2})-\theta(0)$: Since
$L_{s_{1},s_{2}}\mid_{[0,\frac{1}{2}]}=R_{s_{1}}\subset\mathbb{R}_{\leq
0}\times\mathbb{R}$ (from Lemma 5.8),
$\|\theta(\frac{1}{2})-\theta(0)\|\not\geq 1$, because that would require
$L_{s_{1},s_{2}}$ to pass through all 4 quadrants.

Then, as $R_{s_{2}}$ goes from $(-1,0)$ to $(0,-1)$,
$\theta(\frac{1}{2})-\theta(0)$ can either be $\frac{\pi}{2}$ or
$-\frac{3\pi}{2}$. However $\theta(\frac{1}{2})-\theta(0)=-\frac{3\pi}{2}$,
only if $L_{s_{1},s_{2}}\mid_{[0,\frac{1}{2}]}$ passes through
$\mathbb{R}_{>0}\times\mathbb{R}$, in contradiction to Lemma 5.8. Therefore,
$\theta(\frac{1}{2})-\theta(0)\neq-\frac{3\pi}{2}$, and thus
$\theta(\frac{1}{2})-\theta(0)=\frac{\pi}{2}$. See Figure 11(a) for a diagram
illustrating this.

1.000000 -1.000000 (20.269500,12.830500)
(23.500000,21.150000)(22.700000,1.100000)(24.818959,8.681165)
(25.000000,9.150000)
(26.000000,11.600000)(28.944500,18.205500)(25.723961,17.639730) 2 -2$(-1,0)$2
-2$(0,-1)$2 -2$L_{s_{1},s_{2}}\mid_{[0,\frac{1}{2}]}=R_{s_{1}}$

(a) $\theta(\frac{1}{2})-\theta(0)=-\frac{3\pi}{2}$ only if
$L_{s_{1},s_{2}}\mid_{[0,\frac{1}{2}]}$ passes through
$\mathbb{R}_{>0}\times\mathbb{R}$.

1.000000 -1.000000 (25.662900,15.790200)
(37.550000,6.700000)(12.550000,22.350000)(20.478664,11.650717) 2 -2$(-1,0)$2
-2$(0,-1)$2 -2$L_{s_{1},s_{2}}\mid_{[\frac{1}{2},1]}=\bar{R}_{s_{2}}$

(b) $\theta(1)-\theta(\frac{1}{2})=-\frac{\pi}{2}$ only if
$L_{s_{1},s_{2}}\mid_{[\frac{1}{2},1]}$ passes through
$\mathbb{R}_{<0}\times\mathbb{R}_{<0}$.

  * •

$\theta(1)-\theta(\frac{1}{2})$: Since
$L_{s_{1},s_{2}}\mid_{[\frac{1}{2},1]}=\bar{R}_{s_{2}}\not\subset\mathbb{R}_{<0}\times\mathbb{R}_{<0}$
(from Lemma 5.9), $\|\theta(1)-\theta(\frac{1}{2})\|\not\geq 1$, because that
would again require $L_{s_{1},s_{2}}\mid_{[0,\frac{1}{2}]}$ to pass through
all 4 quadrants.

Then, as $L_{s_{1},s_{2}}\mid_{[0,\frac{1}{2}]}$ goes from $(0,-1)$ to
$(-1,0)$, $\theta(1)-\theta(\frac{1}{2})$ can either be $\frac{\pi}{2}$ or
$-\frac{3\pi}{2}$. However, $\theta(1)-\theta(\frac{1}{2})=-\frac{\pi}{2}$
only if $L_{s_{1},s_{2}}\mid_{[\frac{1}{2},1]}$ travels through
$\mathbb{R}_{<0}\times\mathbb{R}_{<0}$, in contradiction to Lemma 5.9.
Therefore, $\theta(1)-\theta(\frac{1}{2})\neq-\frac{\pi}{2}$, and thus
$\theta(1)-\theta(\frac{1}{2})=\frac{3\pi}{2}$. See Figure 11(b) for a diagram
illustrating this.

Hence,

$\eta(L_{s_{1},s_{2}})=\frac{\theta(1)-\theta(0)}{2\pi}=\frac{(\theta(1)-\theta(\frac{1}{2}))+(\theta(\frac{1}{2})-\theta(0))}{2\pi}=\frac{\frac{\pi}{2}+\frac{3\pi}{2}}{2\pi}=1.$

Then, as discussed above, it follows that $L_{s_{1},s_{2}}\not\simeq
c^{\prime\prime}$ in $X^{\prime}$. ∎

We now conclude with the proof of Theorem 5.2.

###### Proof of Theorem 5.2.

If $R_{s}$ doesn’t contains $(0,0)$ for any $s\in(0,1)$, then Lemmata 5.7 and
5.10 tell us respectively that $L_{s_{1},s_{2}}\simeq c^{\prime\prime}$ in
$X^{\prime}$ and that $L_{s_{1},s_{2}}\not\simeq c^{\prime\prime}$, which is a
contradiction. Therefore, our assumption for these lemmata is false, so there
exists some $s^{\prime}$ and $t^{\prime}$ such that
$R_{s^{\prime}}(t^{\prime})=(0,0)$. This gives us two points—namely,
$\gamma(s^{\prime})$ and $\gamma(t^{\prime})$—that, by the construction of
$R_{s}$, inscribe an equilateral triangle such that one of its vertices is
$o$. ∎

## Acknowledgements

We would like to thank Rachana Madhukara for helpful comments.

## References

  * [AK13] Arseniy Akopyan and Roman Karasev.  Inscribing a regular octahedron into polytopes, 2013.
  * [Emc16] Arnold Emch.  On some properties of the medians of closed continuous curves formed by analytic arcs.  Amer. J. Math., 38(1):6–18, 1916.
  * [Mak16] V. V. Makeev.  Planar sections of three-dimensional cylinders.  Vestnik St. Petersburg Univ. Math., 49(4):359–360, 2016.
  * [Mat14] Benjamin Matschke.  A survey on the square peg problem.  Notices Amer. Math. Soc., 61(4):346–352, 2014.
  * [Mey80] Mark D. Meyerson.  Equilateral triangles and continuous curves.  Fund. Math., 110(1):1–9, 1980.
  * [Mey81] Mark D. Meyerson.  Balancing acts.  Topology Proc., 6(1):59–75 (1982), 1981.
  * [Nie92] Mark J. Nielsen.  Triangles inscribed in simple closed curves.  Geom. Dedicata, 43(3):291–297, 1992.
  * [NW95] Mark J. Nielsen and S. E. Wright.  Rectangles inscribed in symmetric continua.  Geom. Dedicata, 56(3):285–297, 1995.
  * [Pak10] Igor Pak.  Lectures on discrete and polyhedral geometry.  http://www.math.ucla.edu/~pak/book.htm, 2010.
  * [Str89] Walter Stromquist.  Inscribed squares and square-like quadrilaterals in closed curves.  Mathematika, 36(2):187–197 (1990), 1989.