2007 IMO Shortlist G5

Let $A B C$ be a fixed triangle, and let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $B C, C A, A B$, respectively. Let $P$ be a variable point on the circumcircle. Let lines $P A_{1}, P B_{1}, P C_{1}$ meet the circumcircle again at $A^{\prime}, B^{\prime}, C^{\prime}$ respectively. Assume that the points $A, B, C, A^{\prime}, B^{\prime}, C^{\prime}$ are distinct, and lines $A A^{\prime}, B B^{\prime}, C C^{\prime}$ form a triangle. Prove that the area of this triangle does not depend on $P$. (United Kingdom)

设$A B C$为固定三角形，$A_{1}、B_{1}、C_{1}$分别为边$B C、C A、A B$的中点。令 $P$ 为外接圆上的变量点。令线 $P A_{1}、P B_{1}、P C_{1}$ 分别在 $A^{\prime}、B^{\prime}、C^{\prime}$ 处再次与外接圆相交。假设点 $A, B, C, A^{\prime}, B^{\prime}, C^{\prime}$ 不同，并且线 $A A^{\prime}, B B^{\prime}, C C^{\prime}$ 形成一个三角形。证明这个三角形的面积不依赖于$P$。 (英国)