Points of squares from triangle sides on circle

Question

Taking a course on geometry, got this problem in my problem set.

Suppose we have a triangle ABC and we take squares $BCP_1P_2$ and $ACP_3P_4$ such that $P_1, P_2, P_3, P_4$ are all on the same circle. How many positive integer triples of angles

Answer 1

The center of the circle on which $P_1,P_2,P_3,P_4$ are is the intersection point between the perpendicular bisector of the line segment $P_1P_2$ and the perpendicular bisector of the line segment $P_3P_4$, i.e. the intersection point between the perpendicular bisector of the side $BC$ and the perpendicular bisector of the side $AC$.

So, we know that the center is the circumcenter and also the orthocenter of $\triangle{ABC}$.

Thus, we know that $\triangle{ABC}$ is an equilateral triangle. (why?)