Elliptic-billiard invariant k_{610}

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix the ellipse $E:x^2/a^2+y^2/b^2=1$, where $a>b>0$, with center $O=(0,0)$ and foci $f_1,f_2=(\pm\sqrt{a^2-b^2},0)$. Fix a confocal elliptic caustic inside $E$ admitting a Poncelet family of $N$-periodic billiard polygons $P=(P_1,\ldots,P_N)$, with $N\geq3$. Their vertices lie on $E$, their sides are tangent to the caustic, and the incoming and outgoing sides obey the billiard reflection law. Indices are cyclic. The outer polygon $P'$ has vertices given by intersections of consecutive tangent lines to $E$ at the $P_i$; the inner polygon $P''$ has vertices at the contacts of the sides of $P$ with the caustic. All constancy assertions concern variation within this fixed family. For any ordered polygon $R=(R_i)_{i=1}^N$, use the signed area \[S(R)=\frac12\sum_{i=1}^N R_i\times R_{i+1},\qquad (x,y)\times(u,v)=xv-yu.\] Write $A=S(P)$, $A'=S(P')$ and $A''=S(P'')$. A pedal polygon $\operatorname{ped}_M(R)$ has vertices the perpendicular feet from $M$ to the lines containing the sides of $R$. An antipedal polygon $\operatorname{ant}_M(R)$ has vertices at consecutive intersections of the lines through $R_i$ perpendicular to $R_i-M$. Define \[A_M=S(\operatorname{ped}_M(P)),\quad A'_M=S(\operatorname{ped}_M(P')),\quad A_M^*=S(\operatorname{ant}_M(P)).\] Primes or double primes on analogous quantities refer to $P'$ or $P''$. For $R$, its vertex centroid and signed area centroid are \[C_0(R)=\frac1N\sum_i R_i,\qquad C_2(R)=\frac{1}{6S(R)}\sum_i(R_i\times R_{i+1})(R_i+R_{i+1}).\] Constructions and quotients are understood wherever their defining intersections and denominators exist. For $j=1,2$, let $Q_{j,i}$ and $Q^*_{j,i}$ be the vertices of $\operatorname{ped}_{f_j}(P)$ and $\operatorname{ant}_{f_j}(P)$, respectively. Put $q_{j,i}=|Q_{j,i}-f_j|$ and $q^*_{j,i}=|Q^*_{j,i}-f_j|$. Let $\bar A_j$, $\bar A'_j$, $\bar A''_j$ be the signed areas of the pedals of $P$, $P'$, $P''$ at $f_j$, and let $\bar A_j^*$, $(\bar A'_j)^*$, $(\bar A''_j)^*$ denote their antipedal signed areas.

Statement recorded in the supplied source.

Prove that for even $N$, $(\bar A''_1)^*/(\bar A''_2)^*$ equals $1$ throughout the family (invariant $k_{610}$). [S2]

Short English statement

For the stated periods, prove that the ratio of the two inner focal-antipedal signed areas equals one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5100038 (AMR-050-0038). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Dan Reznik, Ronaldo Garcia and Jair Koiller, Eighty New Invariants in the Elliptic Billiard, arXiv:2004.12497v11 (29 October 2020), §3.7 and Table 7, pp. 7--9. https://arxiv.org/abs/2004.12497v11.