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{\bf\large On the Poncelet theorem}
{\bf G.Chelnokov, E.Diomidov, V.Kalashnikov, P.Kozhevnikov, A.Zaslavsky}
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The simplest formulation of the Poncelet theorem is next.
{\bf Poncelet theorem.} Let two circles be given, and one of them lies inside the second one. A tangent from an arbitrary point $A_0$ of the external circle $\Omega$ to the internal circle $\omega$ meets $\Omega$ for the second time at point $A_1$. Similarly from point $A_1$ construct point $A_2$ etc. Then if $A_0=A_n$ for some point $A_0$, this ill be true for any other point of $\Omega$.
Not formally an inscribed and circumscribed polygon\footnote{we will use this term in place of closed broken line, which may be self-intersecting} can be "rotated"\, between two circles (its form can change during this rotation). We will call such "rotating"\, polygon {\it a Poncelet polygon}.
This project purposes to prove the Poncelet theorem and to examine some properties of Poncelet polygons. Also we consider the generalizations of Poncelet theorem and some similar theorems.
\section{Poncelet theorem for $n=3, 4$}
\ \par\vskip-\baselineskip
\zd Let $O$, $I$ be the circumcenter and the incenter of the triangle, and $R$, $r$ be the radii of the circumcircle and the incircle respectively. Prove {\bf the Euler formula}
$$
OI^2=R^2-2Rr.
$$
\zd Prove the Poncelet theorem for $n=3$.
A set of remarkable points or centers is associated with any triangle. When we "rotate"\, a triangle between its circumcircle and incircle, these points move on some curves. In the next problems we have to find the corresponding trajectories.
\zd Find the trajectory of
\pp the centroid $M$;
\pp the orthocenter $H$;
\pp \hspace{-4mm}$^*$ the Gergonne point $G$ (the common point of the segments between the vertices of the triangle and the touching points of the opposite sides with the incircle)
\pp \hspace{-3mm}$^*$ the Lemoine point $L$, isogonally conjugated to $M$
of the Poncelet triangle?
\zd Let $A'$, $B'$, $C'$ be the touching points of the incircle with the sides. Find the trajectory of the centroid $M_0$ of triangle $A'B'C'$.
\zd \hspace{-3mm}$^*$ Let a Poncelet triangle and a fixed point $P$ be given. Find the trajectory of the point isogonally conjugated to $P$.
\zd \hspace{-4mm}$^*$ Let $X$ be a fixed point of $\Omega$. Prove that its Simson line passes through a fixed point $Y$ (and line $\ell$, passing through $Y$ and perpendicular to $XY$, touches $\omega$).
\zd A circle touching sides $AC$, $BC$ and the circumcircle of triangle $ABC$ is called {\it semi-inscribed} circle of this triangle.
\pp Find the trajectory of the center of the semi-inscribed circle.
\pp Prove that the semi-inscribed circle of the Poncelet triangle touches some circle distinct from $\Omega$.
\pp Prove the same assertion for the circle, passing through two vertices of the Poncelet triangle and touching $\omega$.
\zd \hspace{-3mm}$^*$ Let triangle $ABC$ and point $X$ be given. Lines $AX$, $BX$, $CX$ meet $BC$, $CA$, $AB$ respectively at points $A'$, $B'$, $C'$. Then the common points of lines $A'B'$ and $AB$, $B'C'$ and $BC$, $C'A'$ and $CA$ are collinear on the line which is called {\it a trypolar} of $X$ wrt $ABC$.
\pp Prove that the trypolar of a fixed point $X$ of $\Omega$ wrt the Poncelet triangle passes through a fixed point $Y$.
\pp Find a locus of points $Y(X)$.
\zd A circle with center $I$ lies inside an other circle. Find the locus of the circumcenters of triangles $IAB$, where $AB$ is an arbitrary chord of the external circle touching the internal one.
\zd Let two circles be given and one of them lies inside the second one. Find the locus of the incenters of triangles $ABC$, where $AC$ and $BC$ are two chords of the external circle touching the internal one.
\zd Two circles with radii 1 meet at two point, and the distance between these points also is equal to 1. $C$ is an arbitrary point on one of these circles, the tangents $CA$ and $CB$ to the second circle meet the first circle for the second time at points $B'$ and $A'$. Find the distance $AA'$.
\zd Let a circle and a point $P$ inside it be given. Two perpendicular rays with origin $P$ meet the circle at points $A$ and $B$.
\pp Find the locus of the midpoints of segments $AB$.
\pp Find the locus of the common points of the tangents to the circle at points $A$ and $B$.
\zd Prove the Poncelet theorem for $n=4$.
\zd Let two circles with centers $O$, $I$ and radii $R$, $r$ satisfy the Poncelet theorem for $n=4$. Find the relation between $R$, $r$ and $d=OI$.
\zd
\pp Prove that the diagonals of the Poncelet quadrilateral meet on the same point $P$, lying on $OI$.
\pp Find the relation between $OP$, $R$ and $d$.
\zd Prove that the lines joining the touching points of the opposite sides of the Poncelet quadrilateral with the incircle are the bisetors of the angles formed by its diagonals.
\zd Find the trajectory of the centroid $M$ of the Poncelet quadrilateral.
\zd \hspace{-3mm}$^*$ Prove that
\pp the product of the tangents of the angles between line $OI$ and the diagonals;
\pp the product of the lengths of the diagonals
of the Poncelet quadrilateral is constant.
\eject
\section{An algebraic view on the Poncelet theorem}
\ \par\vskip-\baselineskip
When $n>4$ the Poncelet theorem also can be proved synthetically. By the examination of the properties of the Poncelet polygons using only geometric methods is difficult. The methods of algebraic geometry are more effective. Firstly prove the Poncelet theorem using these methods.
Consider center $O$ of $\Omega$ as an origin of a coordinates system and line $OI$ as an abscissa axis. Let $R$, $r$ --- be the radii of the circles, and $d$ be the distance between its centers, i.e. $I$ has the coordinates $(d,0)$. Define the coordinates of the points of $\Omega$ as
$x=R(1-t^2)/(1+t^2), y=R \cdot 2t/(1+t^2)$, this concordance between the points of $\Omega$ and the significances of $t$ will be one-one, if point $(-R,0)$ correspond to $t=\infty$. Such defining of a curve is called its rational parametrization. Let $t_0, t_1, \ldots, t_{n-1}$ be the significances of $t$, corresponding to the vertices of the polygon.
\zd \pp Find the relation between $t_0$ and $t_1$.
\pp Find the relation between $t_0$ and $t_2$.
\pp \hspace{-3mm}$^*$ Prove that $t_0$ and $t_n$ satisfy to the relation $P_n(t_0,t_n)=0$, where $P_n(x,y)$ is some symmetric polynomial, having degree 2 on each variable.
\zd Prove the Poncelet theorem.
{\bf The general Poncelet theorem.} Let circles $\omega_1,\ldots,\omega_n$ lie inside circle $\Omega$, and let all these circles be coaxial, i.e they have a common radical axis. If there exists a polygon $A_1\ldots A_n$ inscribed into $\Omega$ and such that $A_1A_2$ touches $\omega_1$, $A_2A_3$ touches $\omega_2$,...,$A_nA_1$ touches $\omega_n$, then there exists an infinite set of such polygons.
\zd \pp Prove the general Poncelet theorem.
\pp Prove "the generalest"\, Poncelet theorem, in which the coaxial circles are replaced by the conics passing through four fixed points.
>From the general Poncelet theorem we obtain that if $A_1\ldots A_n$ is the Poncelet polygon inscribed into circle $\Omega$ and circumscribed around circle $\omega$, then its diagonals $A_iA_{i+k}$ for any fixed $k$ touche the same circle coaxial with $\Omega$ and $\omega$.
\zd Let $R$ and $r$ be the radii of the circumcircle and the incircle of the Poncelet polygon , and $d$ be the distance between its centers. Find the radius of the circle touching the diagonals $A_iA_{i+2}$ and the distance from the center of this circle to the circumcenter.
\zd Find the relations between $R$, $r$ and $d$ for the Poncelet
\pp hexagon;
\pp octagon;
\pp pentagon.
\zd (S.Markelov) Let $R$, $r$ and $d$ be the radii of the circumcircle and the incircle of the Poncelet $n$-gon and the distance between its centers. Prove that $d$, $r$ and $R$ are also the radii of the circumcircle an the incircle and th distance between its centers for some Poncelet polygon, having $n$, $2n$, or $n/2$ sides.
\zd Find the trajectory
\pp the centroid of the vertices;
\pp the centroid of the touching points of the incircle with the sides
of the Poncelet polygon.
\zd Let an incenter, a circumcenter and a centroid of an inscribed and circumscribed $n$-gon be given. Can this $n$-gon be restored by a compass and a ruler if
\pp $n=3$?
\pp $n=4$?
\zd
\pp Let $t_1,\ldots,t_n$ be the values of the parameter corresponding to the vertices $A_1,\ldots,A_n$ of a Poncelet $n$-gon; let $\sigma_1=t_1+\cdots+t_n$, $\sigma_2=t_1t_2+t_1t_3+\cdots+t_{n-1}t_n$,\ldots,$\sigma_n=t_1\cdots t_n$ be the Vieta polynomials from $t_1,\ldots,t_n$. Prove that all even Vieta polynomials are constant and all odd polynomials are proportional to $\sigma_1$.
\pp Let $d_1,\ldots,d_n$ be the lengths of the tangents from the vertices $A_1,\ldots,A_n$ of a Poncelet $n$-gon to its incircle; let $\sigma_1=d_1+\cdots+d_n$, $\sigma_2=d_1d_2+d_1d_3+\cdots+d_{n-1}d_n$,\ldots,$\sigma_n=d_1\cdots d_n$ be the Vieta polynomials from $d_1,\ldots,d_n$. Prove that all even Vieta polynomials are constant and all odd polynomials are proportional to $\sigma_1$.
\zd \hspace{-3mm}$^*$ Let two circle be given one of them lying inside the other. Consider a broken lines $A_1A_2\ldots A_{n+1}$, with the vertices lying on the external circle, and the links touching the internal one. Find the locus of the centroids of the touching points.
\zd \hspace{-3mm}$^*$ Define the Simson line of point $X$ wrt a cyclic $n$-gon using the induction as the line containing the projections of $X$ to the Simson lines of $X$ wrt $(n-1)$-gons, obtaining by deleting of each vertex. Prove that the Simson line of a fixed point $Х$ of $\Omega$ wrt the Poncelet polygon passes through a fixed point.
\zd Let triangle $ABC$ is inscribed into circle $\Omega$ with radius 1, and let lines $AB$, $BC$, $CA$ touches circles $\omega_1$, $\omega_2$, $\omega_3$, in such a way that all these circles are coaxial. Find the relation between the distances $d_1$, $d_2$, $d_3$ from the centers of $\omega_1$, $\omega_2$, $\omega_3$ to the center of$\Omega$.
\section{The other closing theorems}
\ \par\vskip-\baselineskip
The Poncelet theorem is the example of a closing theorem. Take some other examples of such theorems.
{\bf The Steiner porism.} Let two circles be given: $\alpha$ and lying inside it $\beta$. Consider a chain of circles $\omega_1,\omega_2,\ldots$, touching $\alpha$ internally, touching $\beta$ externally, and such that $\omega_{i+1}$ touches $\omega_i$. If for some circle $\omega_1$ circle $\omega_n$ touches $\omega_1$, then this is true for any $\omega_1$.
{\bf The zigzag theorem.} Let two circles $\alpha$ and $\beta$ be given. Take an arbitrary point $A_0$ on $\alpha$ and find such point $B_0$ on $\beta$ that $A_0B_0=1$. Now find point $A_1$ on $\alpha$ distinct from $A_0$ and such that $A_1B_0=1$ etc. If $A_n$ coincides with $A_0$, then this is true for any other point $A_0$.
Note that the zigzag theorem is correct even for two circles not lying in the same plane.
{\bf The Emch theorem.} Let three circles be given: $\alpha$, $\beta$ lying inside $\alpha$ and $\gamma$ lying inside $\beta$. Consider a chain of circles $\omega_1,\omega_2,\ldots$, touching $\alpha$ internally, touching $\gamma$ externally, and such that $\omega_{i+1}$ and $\omega_i$ meet at the point lying on $\beta$. If for some $\omega_1$ circle $\omega_n$ touches $\omega_1$, then this is true for any $\omega_1$.
{\bf The Brocard broken line theorem.} Let a circle $\omega$, a point $P$ inside it and an angle $\phi$ be given. For an arbitrary point $X_0$ of $\omega$ construct such point $X_1$, that $\angle PX_0X_1=\phi$. Similarly for point $X_1$ construct $X_2$ etc. If for some $X_0$ $X_n=X_0$, then this is true for any $X_0$.
{\bf The Protasov theorem.} Let $S_0$, $S_1$, $S_2$ be three spheres with non-collinear centers. Consider a family $\Sigma$ of spheres touching $S_1$ and $S_2$ (the spheres of $\Sigma$ touche each of spheres $S_1$, $S_2$ by the same way --- internally or externally) and perpendicular to $S_0$. Let $\omega$~--- be a circle in the space, which do not lie on any sphere from $\Sigma$ and do not pass through the common points of many than two spheres of $\Sigma$. For an arbitrary point $X_0$ on $\omega$ take a sphere $s_1\in\Sigma$ passing through it and find the second common point $X_1$ of $s_1$ and $\omega$. Take a sphere $s_2\in\Sigma$ distinct from $s_1$ and passing through $X_1$ and find its second common point $X_2$ with $\omega$ etc. If for some $X_0$ $X_n=X_0$, then this is true for any $X_0$.
\zd Prove these theorems algebraically.
An arbitrary circle on the plane can be given by an equation $x^2+y^2+ax+by+c=0$. Correspond to such circle a point in the space with coordinates $(a,b,c)$.
\zd Is this a one-one correspondence?
\zd Which pairs of points correspond to touching circles?
\zd Which assertions correspond to the Steiner and Emch theorems?
\zd Obtain
\pp the Emch theorem and the Brocard broken line theorem from the Poncelet theorem;
\pp the Poncelet theorem, the zigzag theorem and the Steiner porism from the Emch theorem.
\zd Find the Protasov theorem
\zd Obtain the zigzag theorem, the Poncelet theorem, the Emch theorem and the Steiner porism from the Protasov theorem.
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