Question

Prove that the quadrilateral formed by internal andlebisectors of any quadrilateral is cyclic

Answer 1

given:- a cyclic quadrilateral ABCD , where AH,DH,BF, and CF are the bisectors of /_A,/_B,/_C and /_ D
such that quadrilateral EFGH is formed

To prove:- EFGH is a cyclic quadrilateral

Proof:- in ️s BFC and AHD
angle BFC+angle FBC+angle FCB= 180°
and angle AHD+angle HAD+angle HDA =180°
=> angle BFC +1/2 angle B+1/2 angle C= 180° ......(1)
and, angle AHD+ 1/2angle A+1/2 angle D=180° .......(2)
adding 1 and 2 we get
angle BFC+ 1/2 angleB +1/2 angle C + 1/2 angle A + 1/2 angle D + angle AHD= 360°
angle BFC + angle AHD + 1/2{( angle A + angle C)+( angle b + angle d)}= 360°
angle BFC + angle AHD +1/2 (180° + 180° )= 360°
Angle BFC + angle AHD=180°
hence proved that E F G H is a cyclic quadrilateral