Question 175
Let $ABC$ be an acute triangle and let $D$ be a point on the side $AB$. The circumcircle of the triangle $BCD$ intersects the side $AC$ at $E$. The circumcircle of the triangle $ADC$ intersects the side $BC$ at $F$. Let $O$ be the circumcentre of the triangle $CEF$. Prove that the points $D$ and $O$ and the circumcentres of the triangles $ADE$, $ADC$, $DBF$ and $DBC$ are concyclic and the line $OD$ is perpendicular to $AB$.