The bisectors of the opposite angles A and C of a cyclic quadrilateral

ABCD intersect the circle at the points X and Y respectively.

Prove that XY is a diameter of the circle.

Given: ABCD is a cyclic quadrilateral. AX and CY are the bisectors of ∠A and ∠C respectively.

To prove: XY is the diameter of the circle i.e. ∠XAY = 90°

Construction: Join AY and YD.

Proof:

ABCD is a cyclic quadrilateral.

∴ ∠A + ∠C = 180°  (Sum of opposite angles of a cyclic quadrilateral is 180° )

⇒ ∠XAD + ∠DCY = 90°    ...(1)  (AX and CY are the bisector of ∠A and ∠C respectively)

∠DCY = ∠DAY ...(2)  (Angles in the same segment are equal)

From (1) and (2), we have

∠XAD + ∠DAY = 90°

⇒ ∠XAY = 90°

⇒ ∠XAY is the angle in a semi-circle.

⇒ XY is the diameter of the circle.