prove the quadrilateral formed by joining the angle bisectors of cycli quadrilateral is a cyclic quadrileteral
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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°.]
arun71:
thank u great job
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