Question 60
Let $O$ be the circumcenter of an acute non-isosceles triangle $ABC$. Lines $BO$ and $CO$ meet sides $AC$ and $AB$ respectively at points $K$ and $N$. Points $P$ and $T$, different from $K$ and $N$, are chosen respectively on $AC$ and $AB$ so that $OK = OP$ and $ON = OT$. A line through $P$ parallel to $BK$ and a line through $T$ parallel to $CN$ meet at point $M$. Prove that the circumradii of triangles $AMB$, $BMC$, $CMA$ are equal.