Question

prove that the quadrilateral formed by the internal angle bisectors of any quadrilateral is cyclic. ​

Answer 1

Heya buddy!!!

Here's the answer to your question...

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: EFGH is a cyclic quadrilateral.

Proof: ∠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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