Question 151
Let $D$ be a point on the side $BC$ of an acute triangle $ABC$. The circle with diameter $BD$ meets the lines $AB$ and $AD$ respectively at the points $X$ and $P$, which are different from the points $B$ and $D$. The circle with diameter $CD$ meets the lines $AC$ and $AD$ respectively at the points $Y$ and $Q$, which are different from the points $C$ and $D$. Through the point $A$ draw two lines which are perpendicular to $PX$ and $QY$ with the feet of perpendicular $M$ and $N$ respectively.
Prove that $\triangle AMN \sim \triangle ABC$ if and only if the line $AD$ passes through the circumcenter of $\triangle ABC$. (Posed by Li Qiusheng)
Prove that $\triangle AMN \sim \triangle ABC$ if and only if the line $AD$ passes through the circumcenter of $\triangle ABC$. (Posed by Li Qiusheng)