$O$ is the center of a circle with $AB$ diameter. If $OD$ is perpendicular to $AB$ and meets chord $AC$ or $AC$ produced at $D$, then prove that circle $AOD$ is equal to the circle through $O,B,D,C$.
The triangles $AOD$ and $ACB$ are similar. I've tried using similarity for the above theorem but we need to show that the two radiuses are equal. Since, $AOD$ is a right-angled triangle, its circumcenter lies on the hypotenuse.
I am not able to proceed further. Please help.