Question 39
Let $\triangle ABC$ be a scalene triangle, and let $D$ and $E$ be points on sides $AB$ and $AC$ respectively such that the circumcircles of triangles $ACD$ and $ABE$ are tangent to $BC$. Let $F$ be the intersection point of $BC$ and $DE$. Prove that $AF$ is perpendicular to the Euler line of $ABC$.
(Recall that the Euler line of a triangle is the line passing through its circumcentre and the orthocentre.)
(Recall that the Euler line of a triangle is the line passing through its circumcentre and the orthocentre.)