If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

In any triangle $ABC$, if angle bisector of $\angle A$ and perpendicular bisector of $BC$ intersect, prove that they intersect on the circumcircle of the triangle $ABC$.