Problem

Source: VNTST 2020

Tags: geometry, circumcircle



In acute $\triangle ABC$, $O$ is the circumcenter, $I$ is the incenter. The incircle touches $BC,CA,AB$ at $D,E,F$. And the points $K,M,N$ are the midpoints of $BC,CA,AB$ respectively. a) Prove that the lines passing through $D,E,F$ in parallel with $IK,IM,IN$ respectively are concurrent. b) Points $T,P,Q$ are the middle points of the major arc $BC,CA,AB$ on $\odot ABC$. Prove that the lines passing through $D,E,F$ in parallel with $IT,IP,IQ$ respectively are concurrent.