Bisectors of angle A & C of a cyclic quadrilateral ABCD intersect a circle through A,B,C,D at E&F respectively. Proove that EF is the diameter of the circle.

**Given: **ABCD is a cyclic quadrilateral. AE and CF are the bisectors of ∠A and ∠C respectively.

**To prove:** EF is the diameter of the circle i.e. ∠EAF = 90°

**Construction:** Join AF and FD.

**Proof:**

ABCD is a cyclic quadrilateral.

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

⇒ ∠EAD + ∠DCF = 90° ...(1) (AE and CF are the bisector of ∠A and ∠C respectively)

∠DCF = ∠DAF ...(2) (Angles in the same segment are equal)

From (1) and (2), we have

∠EAD + ∠DAF = 90°

⇒ ∠EAF = 90°

⇒ ∠EAF is the angle in a semi-circle.

⇒ EF is the diameter of the circle.

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