Question 3
Let $ABC$ be an acute triangle with circumcenter $O$. Let $A'$ be the center of the circle passing through $C$ and tangent to $AB$ at $A$; define $B'$ as the center of the circle through $A$ tangent to $BC$ at $B$, and $C'$ as the center of the circle through $B$ tangent to $CA$ at $C$.
(a) Prove that the area of $A'B'C'$ is at least the area of $ABC$.
(b) Let $X,Y,Z$ be the perpendicular projections of $O$ onto $A'B',B'C',C'A'$, respectively. Suppose the circumcircle of $XYZ$ meets these three lines again at $X',Y',Z'$, respectively, with $X'\ne X,Y'\ne Y,Z'\ne Z$. Prove that $AX',BY',CZ'$ are concurrent.
(a) Prove that the area of $A'B'C'$ is at least the area of $ABC$.
(b) Let $X,Y,Z$ be the perpendicular projections of $O$ onto $A'B',B'C',C'A'$, respectively. Suppose the circumcircle of $XYZ$ meets these three lines again at $X',Y',Z'$, respectively, with $X'\ne X,Y'\ne Y,Z'\ne Z$. Prove that $AX',BY',CZ'$ are concurrent.