Question

The bisectors of the opposite angles A and C of a cydic quadrilateral ABCD intersect the cirde at the points E and F, respectively. Prove that EF is a diameter of the circle.

Answer 1

In cyclic quadrilateral ABCD

∠A + ∠ C = 180°

1/2 ∠A + 1/2 ∠C = 90°

∠EAB + ∠BCF = 90°   -(1) (AE bisects ∠ A ; CF bisects ∠C)

Also ,

∠BCF = ∠BAF  - (2) (Angles in the same segment)

Using (1) in (2) we get ,

∠EAB + ∠BAF = 90°

∠FAE = 90°

EF is the diameter of the circle ,

∴ angle in a semi circle is a right angle.