Problem

Source: Iran Team selection test 2024 - P2

Tags: geometry



For a right angled triangle $\triangle ABC$ with $\angle A=90$ we have $AC=2AB$. Point $M$ is the midpoint of side $BC$ and $I$ is incenter of triangle $\triangle ABC$. The line passing trough $M$ and perpendicular to $BI$ intersect with lines $BI$ and $AC$ at points $H$ and $K$ respectively. If the semi-line $IK$ cuts circumcircle of triangle $\triangle ABC$ at $F$ and $S$ be the second intersection point of line $FH$ with circumcircle of triangle $\triangle ABC$ , then prove that $SM$ is tangent to the incircle of triangle $\triangle ABC$. Proposed by Mahdi Etesami Fard