Hope everyone else is doing loads of maths during the lockdown like me.can’t quite rap my head around this geometry question.
Problem: Consider the acute-angled triangle ABC. Choose the points M and N on the sides AB and AC respectively such that the distance from M to line BC equals |AM|, and the distance from N to BC equals |AN|. The perpendicular line from A to AB intersects the line BC at the point R, and the perpendicular line from A to AC intersects the line BC at the point S. Let I be the intersection of MR and NS. Prove that AI is the angle bisector of the angle SAR?
Any help would be much appreciated. Stay safe.
$\triangle ASN \cong \triangle A_2SN$ implies S(I)N is the angle bisector of $\angle ASR$.
Similarly R(I)M is the angle bisector of $\angle ARS$. That means I is the in-center of $\triangle ASR$. Result follows.