Triangle bases altures d'un altre triangle
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.
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.
Resulta que el tringle inscrit és el triangle inscrit amb el perímetre més petit (teorema de Fagnano).
Està basat en aquest post de John Baez a Bluesky.
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.)
But what about obtuse triangles?
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°.
But what about beyond that?
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: 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.]
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.
If you're a serious mathematician and billiards sound too frivolous, good news:
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!