Donat un triangle acutangle, si dibuixem un triangle unint les bases de les altures, les bisectrius d'aquest nou triangle coincideixen amb les altures del primer triangle.[br][br]I al revés: donat un triangle, si dibuixem el triangle format per les perpendiculars a les bisectrius pels vèrtexs del triangle, les altures del nou triangle, coincideixen amb les bisectrius del primer triangle.

Some billiard news! To set the stage: in 1775, Giovanni Fagnagno proved that for every acute triangle, there's a way to bounce a billiard ball inside in a periodic orbit - an orbit that repeats itself. (Read the alt text for the trick.)[br][br]But what about obtuse triangles?[br][br]Obtuse triangles are much harder. Rich Schwartz's computer-assisted "McBilliards" project searched over combinatorial "orbit types" and proved that every obtuse triangle with largest angle up to 100° has a periodic orbit. Later work pushed the bound to 112.3°. [br][br]But what about beyond that?[br][br]Now Giovanni Forno claims he sank it in the pocket and proved *every* polygon admits a periodic billiard orbit! [https://arxiv.org/abs/2606.10102: [i]We prove that the billiard flow in any finite polygon has at least one periodic orbit. The proof by contradiction is based on a fundamental result on the dynamics of the billiard flow by Galperin, Krüger and Troubetzkoy, on the geometry of a one-parameter scaling of the natural Riemannian metric on the unit tangent bundle, and on the topology of the skeleton or cut-locus of the scaled metrics[/i].][br][br]But I haven't checked his proof. And it's nonconstructive, so there's still room to develop algorithms to find period orbits - even for triangular billiard tables.[br][br]If you're a serious mathematician and billiards sound too frivolous, good news: [br][br]Forno claims to derive his result from something more profound. Every compact 2-manifold with a flat Riemannian metric with finitely many conical singularities has a periodic geodesic!