Problem

Source: 2010 Oral Moscow Geometry Olympiad grades 8-9 p4

Tags: geometry, triangle inequality, right triangle, isosceles



An isosceles triangle $ABC$ with base $AC$ is given. Point $H$ is the intersection of altitudes. On the sides $AB$ and $BC$, points $M$ and $K$ are selected, respectively, so that the angle $KMH$ is right. Prove that a right-angled triangle can be constructed from the segments $AK, CM$ and $MK$.