OIM 2008 Problem 5

Let $ABC$ a triangle and $X$ , $Y$ and $Z$ points at the segments $BC$ , $AC$ and $AB$ , respectively.Let $A'$ , $B'$ and $C'$ the circuncenters of triangles $AZY$ , $BXZ$ , $CYX$ , respectively.Prove that $4(A'B'C')\geq(ABC)$ with equality if and only if $AA'$ , $BB'$ and $CC'$ are concurrents.

Note: $(XYZ)$ denotes the area of $XYZ$