Question 88
Let $A_1$, $B_1$, $C_1$ be midpoints of sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$ respectively. Let $C_1L$ and $C_1K$ be bisectrices of $\triangle ACC_1$ and $\triangle BCC_1$ respectively. The line $C_1B_1$ intersects the line $LK$ at $A_2$, and the line $C_1A_1$ intersects the line $LK$ at $B_2$.
Prove that the lines $AA_2$, $BB_2$, $CC_1$ are concurrent.
Prove that the lines $AA_2$, $BB_2$, $CC_1$ are concurrent.