Question

Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic

Answer 1

ABCD is a cyclic quadrilateral
∠A + ∠C = 180° and ∠B + ∠D = 180°
(∠A + ∠C)/2 = 90° and (∠B + ∠D)/2 = 90°
w,x,y,z are angles of the inner quadrilateral
x + z = 90° and y + w = 90°
In ΔAGD and ΔBEC,
x + y + ∠AGD = 180° and z + w + ∠BEC = 180°
∠AGD = 180° – (x+y) and ∠BEC = 180° – (z+w)
∠AGD + ∠BEC = 360° – (x+y+z+w) = 360° – (90+90) = 360° – 180° = 180°
∠AGD+∠BEC = 180°
∠FGH+∠HEF = 180°
The sum of a pair of opposite angles of a quadrilateral EFGH is 180°.
Hence EFGH is cyclic

Answer 2

Answer:

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