On the incenters of triangular orbits on elliptic billiards

Abstract

We consider 3-periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. 1. THE STATEMENT OF THE THEOREM AND THE IDEA OF THE PROOF Elliptic billiards are at the same time classical and popular subject (see, for example [1], [2], [3] and [4]) since they continue to deliver interesting problems. We will consider an ellipse and a billiard in it with the standard reflection law : the angle of incidence equals the angle of reflection. Let the trajectory from a point on the boundary repeat itself after two reflections : this means that we obtained a triangle which presents a 3-periodic trajectory of the ball in the elliptic billiard. Poncelet’s famous theorem [5] states that the sides of these triangles are tangent to some smaller ellipse confocal to the initial one. We prove the following fact which was observed experimentally by Dan Reznik [10] : THEOREM 1.1. For every elliptic billiard the set of incenters (the centers of the inscribed circles) of its triangular orbits is an ellipse. ∗ ) Supported in part by RFBR grants 12−01−31241 mol-a and 12−01−33020 mol-a-ved. ar X iv :1 30 4. 75 88 v2 [ m at h. D S] 1 2 N ov 2 01 3