Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Let ABCD be a quadrilateral circumscribing a circle centered at O such that it touches the circle at point P, Q, R, S.
Let us join the vertices of the quadrilateral ABCD to the center of the circle.
AP = AS (Tangents from the same point)
OP = OS (Radii of the same circle)
OA = OA ( common side )
Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.