Question 38
Let $ABC$ be a fixed triangle, and let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC, CA, AB$, respectively. Let $P$ be a variable point on the circumcircle. Let lines $PA_{1}, PB_{1}, PC_{1}$ meet the circumcircle again at $A', B', C'$ respectively. Assume that the points $A, B, C, A', B', C'$ are distinct, and lines $AA', BB', CC'$ form a triangle. Prove that the area of this triangle does not depend on $P$.