CAPS 2018 Problem 2

Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$ , respectively, such that $BD=CE$ . Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$ , respectively. Prove that the circumcircles of the triangles $ABC, ADE$ , and $AO_1O_2$ have a common point different from $A$ .

Proposed by Patrik Bak, Slovakia