We assumed that there exist four points that have equal angles as shown on picture below, and we want to prove that these four points lie on the one circle by contradiction.
We know that inscribed angles that subtend the same arc on the circle are equal. So we are drawing an additional line and get one more green angle that equal previous two:
So now we have a triangle with interior and exterior angles that are equal. Hence, those two points are the same ones. 🙂