Question 180
In acute triangle $ABC$ which is not isosceles, $AP$, $BQ$, $CR$ are three altitudes, $H$ is the orthocenter. The parallel line to $BC$ passing through $A$ intersects line $RQ$ at point $D$. Let $A_1$ be the midpoint of $BC$, and let $K$ be the intersection of $RQ$ and $AA_1$. The line passing through the midpoint of $AH$ and point $K$ intersects line $DA_1$ at point $A_2$. Similarly define points $B_2$ and $C_2$.
Suppose that the circumcircle of non-degenerate triangle $A_2B_2C_2$ is $\omega$. Prove: There exist three circles $\odot A'$, $\odot B'$ and $\odot C'$ inside $\omega$ that are tangent to $\omega$ and satisfy the following conditions:
(1) $\odot A'$ is tangent to sides $AB$, $AC$, $\odot B'$ is tangent to sides $BA$, $BC$, and $\odot C'$ is tangent to sides $CA$, $CB$;
(2) The centers of the three circles, $A'$, $B'$, $C'$, are distinct and collinear.
Suppose that the circumcircle of non-degenerate triangle $A_2B_2C_2$ is $\omega$. Prove: There exist three circles $\odot A'$, $\odot B'$ and $\odot C'$ inside $\omega$ that are tangent to $\omega$ and satisfy the following conditions:
(1) $\odot A'$ is tangent to sides $AB$, $AC$, $\odot B'$ is tangent to sides $BA$, $BC$, and $\odot C'$ is tangent to sides $CA$, $CB$;
(2) The centers of the three circles, $A'$, $B'$, $C'$, are distinct and collinear.