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Hence,Prove that quadrilateral formed by angle bisectors of acyclic quadrilateral is also cyclic.INCERT) |
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Answer» Given a quadrilateral ABCD with internal angle bisectors AF, BH, CHand DF of angles A, B, C and D respectively and the points E, F, Gand H form a quadrilateral EFGH. To prove that EFGH is a cyclic quadrilateral. ∠HEF = ∠AEB [Vertically opposite angles] -------- (1) Consider triangle AEB, ∠AEB + ½ ∠A + ½ ∠ B = 180° ∠AEB = 180° – ½ (∠A + ∠ B) -------- (2) From (1) and (2), ∠HEF = 180° – ½ (∠A + ∠ B) --------- (3) Similarly, ∠HGF = 180° – ½ (∠C + ∠ D) -------- (4) From 3 and 4, ∠HEF + ∠HGF = 360° – ½ (∠A + ∠B + ∠C + ∠ D)= 360° – ½(360°) = 360° – 180° = 180°So, EFGH is a cyclic quadrilateral since the sum of the oppositeangles of the quadrilateral is 180°.] |
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