Question 4
Triangle $ABC$ has orthocenter $H$ and incenter $I$. On rays $AB,AC,BC,BA,CA,CB$, respectively, choose $A_1,A_2,B_1,B_2,C_1,C_2$ so that
$$AA_1=AA_2=BC,\quad BB_1=BB_2=CA,\quad CC_1=CC_2=AB.$$
The pairs of lines $(B_1B_2,C_1C_2)$, $(C_1C_2,A_1A_2)$, $(A_1A_2,B_1B_2)$ meet at $A',B',C'$, respectively.
(a) Prove that the area of $A'B'C'$ is at most the area of $ABC$.
(b) Let $J$ be the circumcenter of $A'B'C'$. Lines $AJ,BJ,CJ$ meet $BC,CA,AB$ at $R,S,T$, respectively. The circumcircles of $AST,BTR,CRS$ have a common point $K$. Prove that if $ABC$ is scalene, then $IHJK$ is a parallelogram.
$$AA_1=AA_2=BC,\quad BB_1=BB_2=CA,\quad CC_1=CC_2=AB.$$
The pairs of lines $(B_1B_2,C_1C_2)$, $(C_1C_2,A_1A_2)$, $(A_1A_2,B_1B_2)$ meet at $A',B',C'$, respectively.
(a) Prove that the area of $A'B'C'$ is at most the area of $ABC$.
(b) Let $J$ be the circumcenter of $A'B'C'$. Lines $AJ,BJ,CJ$ meet $BC,CA,AB$ at $R,S,T$, respectively. The circumcircles of $AST,BTR,CRS$ have a common point $K$. Prove that if $ABC$ is scalene, then $IHJK$ is a parallelogram.