BALTIC-WAY 2022 Problem 11

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$ . The circle with centre on the line $AB$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $D$ . Similarly, the circle with centre on the line $AC$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $E$ . Prove that $BD$ is parallel with $CE$ .