Question 2
Let $ABC$ be an acute scalene triangle inscribed in a circle with centre $O$. Its incircle, with centre $I$, touches $BC,CA,AB$ at $D,E,F$, respectively. Let $K,M,N$ be the midpoints of $BC,CA,AB$, respectively.
(a) Prove that the lines through $D,E,F$ parallel to $IK,IM,IN$, respectively, are concurrent.
(b) Let $T,P,Q$ be the midpoints of the major arcs $BC,CA,AB$, respectively, of the circumcircle. Prove that the lines through $D,E,F$ parallel to $IT,IP,IQ$, respectively, are concurrent.
(a) Prove that the lines through $D,E,F$ parallel to $IK,IM,IN$, respectively, are concurrent.
(b) Let $T,P,Q$ be the midpoints of the major arcs $BC,CA,AB$, respectively, of the circumcircle. Prove that the lines through $D,E,F$ parallel to $IT,IP,IQ$, respectively, are concurrent.