Problem

Source: Indian TST Day 3 Problem 1

Tags: geometry, circumcircle, trigonometry, geometric transformation, reflection, dilation, radical axis



In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.