Question 82
A triangle $ABC$ inscribed in a circle with center $O$ has three altitudes $AH$, $BK$, $CL$. Let $A_0$, $B_0$, $C_0$ respectively be the midpoints of $AH$, $BK$, $CL$.
The incircle with center $I$ of triangle $ABC$ touches the sides $BC$, $CA$, $AB$ respectively at $D$, $E$, $F$.
Prove that the four lines $A_0D$, $B_0E$, $C_0F$ and $OI$ are concurrent. (When $O$ coincides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$).
The incircle with center $I$ of triangle $ABC$ touches the sides $BC$, $CA$, $AB$ respectively at $D$, $E$, $F$.
Prove that the four lines $A_0D$, $B_0E$, $C_0F$ and $OI$ are concurrent. (When $O$ coincides with $I$, we consider the line $OI$ as an arbitrary line passing through $O$).