Problem

Source: 2023 China TST Problem 8

Tags: geometry, China TST



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