2017 APMO 第 2 题

Let $A B C$ be a triangle with $A B<A C$. Let $D$ be the intersection point of the internal bisector of angle $B A C$ and the circumcircle of $A B C$. Let $Z$ be the intersection point of the perpendicular bisector of $A C$ with the external bisector of angle $\angle B A C$. Prove that the midpoint of the segment $A B$ lies on the circumcircle of triangle $A D Z$.

令 $A B C$ 为一个三角形，$A B<A C$。设$D$为角$B A C$的内平分线与$A B C$的外接圆的交点。令$Z$ 为$A C$ 的垂直平分线与角$\angle B A C$ 的外平分线的交点。证明线段$A B$ 的中点位于三角形$A D Z$ 的外接圆上。