2020 IMO Shortlist G1

Let $A B C$ be an isosceles triangle with $B C=C A$, and let $D$ be a point inside side $A B$ such that $A D<D B$. Let $P$ and $Q$ be two points inside sides $B C$ and $C A$, respectively, such that $\angle D P B=\angle D Q A=90^{\circ}$. Let the perpendicular bisector of $P Q$ meet line segment $C Q$ at $E$, and let the circumcircles of triangles $A B C$ and $C P Q$ meet again at point $F$, different from $C$. Suppose that $P, E, F$ are collinear. Prove that $\angle A C B=90^{\circ}$. (Luxembourg)

令$A B C$ 为等腰三角形，且$B C=C A$，并令$D$ 为边$A B$ 内的点，使得$A D<D B$。令$P$和$Q$分别为$B C$和$C A$内侧的两个点，使得$\angle D P B=\angle D Q A=90^{\circ}$。让$P Q$的垂直平分线在$E$处与线段$C Q$相交，并让三角形$A B C$和$C P Q$的外接圆在与$C$不同的点$F$处再次相交。假设$P、E、F$共线。证明$\angle A C B=90^{\circ}$。 (卢森堡)