CN-TST 2023 Problem 8

In non-isosceles acute ${}{\triangle ABC}$ , $AP$ , $BQ$ , $CR$ is the height of the triangle. $A_1$ is the midpoint of $BC$ , $AA_1$ intersects $QR$ at $K$ , $QR$ intersects a straight line that crosses ${A}$ and is parallel to $BC$ at point ${D}$ , the line connecting the midpoint of $AH$ and ${K}$ intersects $DA_1$ at $A_2$ . Similarly define $B_2$ , $C_2$ . ${}\triangle A_2B_2C_2$ is known to be non-degenerate, and its circumscribed circle is $\omega$ . Prove that: there are circles $\odot A'$ , $\odot B'$ , $\odot C'$ tangent to and INSIDE $\omega$ satisfying: (1) $\odot A'$ is tangent to $AB$ and $AC$ , $\odot B'$ is tangent to $BC$ and $BA$ , and $\odot C'$ is tangent to $CA$ and $CB$ . (2) $A'$ , $B'$ , $C$ ' are different and collinear. Created by Sihui Zhang