Question

i will mark it brainliest for first In c(o,r) abcd is cyclic quad. EF is diameter prove AE and CF bisect angle A and angle C respectively

Answer 1

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 AE and FD.

Proof:

ABCD is a cyclic quadrilateral.

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

1/2angleA+1/2angleC=90°

⇒ ∠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.