2021 APMO 第 3 题

Let $A B C D$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals $A C$ and $B D$, let $L$ be the center of the circle tangent to sides $A B, B C$, and $C D$, and let $M$ be the midpoint of the arc $B C$ of $\Gamma$ not containing $A$ and $D$. Prove that the excenter of triangle $B C E$ opposite $E$ lies on the line $L M$.

Let $A B C D$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle.设$E$为对角线$A C$和$B D$的交点，设$L$为与边$A B、B C$和$C D$相切的圆心，设$M$为不包含$A$和$D$的$\Gamma$的弧$B C$的中点。 Prove that the excenter of triangle $B C E$ opposite $E$ lies on the line $L M$.