Question

18 prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.

Answer 1

Given a quadrilateral ABCD with internal angle bisectors AF, BH, CH

and DF of angles A, B, C and D respectively and the points E, F, G

and 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 opposite

angles of the quadrilateral is 180°.

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