the bisectors of the opposite angles A and C of cyclic quadrilateral ABCD intersect the circle at the point E - Maths - Circles

Question

the bisectors of the opposite angles A and C of cyclic
quadrilateral ABCD intersect the circle at the point E and F prove
that EF is a diameter of the circle

Ayushi Sharma · Answer

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 AE and FD Proof: ABCD is a cyclic quadrilateral ∴ ∠A + ∠C = 180° (Sum of opposite angles of a cyclic quadrilateral is 180° ) $n n \Rightarrow n n \frac{n}{n 1 n} n â¢ n n n n ∠ n n n n A + n \frac{n}{n 1 n} n â¢ n n n n ∠ n n n C = 90ï¿½ n n$ ⇒ ∠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

the bisectors of the opposite angles A and C of cyclic quadrilateral ABCD intersect the circle at the point E and F.prove that EF is a diameter of the circle.