Prove that angle bisectors of the angles formed by producing opp sides of a cyclic quadrilateral ( provided they are - Maths - Quadrilaterals

Question

Prove that angle bisectors of the angles formed by producing opp
sides of a cyclic quadrilateral ( provided they are not parallel )
intersect at right angle

Koka Sri Lakshmi Divya Sai · Accepted Answer

$A cyclic quadrilateral ABCD whose opposite sides when produced intersect at the [points P and Q.The bisectors PF and QF of ∠ P and ∠ Q respectively at F ProduceQF to meet AB at E In Δ QDG and Δ QBE ∠ 1 = ∠ 2 (QF is bisector of ∠ DQC) ∠ 3 = ∠ 4 Exterior angles of cyclic quadrilateral=its interior opposite angles ∴ ∠ 5 = ∠ 6 - - - - (i) But ∠ 5 = ∠ 7 (Vertically opposite ang les)-----(ii) ∴ ∠ 6 = ∠ 7 (using (i) and (ii)) Now in Δ PGF and Δ PEF ∠ 9 = ∠ 8 and ∠ 7 = ∠ 6 ∴ ∠ PFG= ∠ PFE But ∠ PFG+ ∠ {PFE=180}^{0} ∴ ∠ PFG= ∠ {PFE=90}^{0} So ∠ {PFQ=90}^{0}$

Prove that angle bisectors of the angles formed by producing opp.sides of a cyclic quadrilateral ( provided they are not parallel ) intersect at right angle .