[Rectangles and Squares. Properties and Characteristics] [Equilateral (Equiangular) Triangle]

A circle constructed on side $A D$ of parallelogram $A B C D$ as a diameter passes through vertex $B$ and the midpoint of side $B C$. Find the angles of the parallelogram.
Let $O$ be the midpoint of side $A D$. Prove that triangle $A B O$ is equilateral.

## Solution

Let $O$ be the midpoint of side $A D$, and $M$ be the midpoint of side $B C$. The circle with center $O$ passes through points $A$, $B$, and $M$, and $B M = A O$ and $B M \| A O$, so $A B = O M = O B = O A$. Therefore, triangle $A B O$ is equilateral. Consequently,

$$
\angle B A D = 60^{\circ}, \angle A B C = 120^{\circ}
$$