Question

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

Answer 1

good afternoon dear....

first see the attachment.

mark the point where bisectors at A and B meet as E and where bisectors at C and D meet as F.

BAE = A/2
ABE = B/2
AEB = 180 - (A/2 + B/2)

like this ,CFD = 180 - (C/2 + D/2)
the sum of opposite angles of the quadrilateral formed

= 180 - (A/2 + B/2) + 180 - (C/2+ D/2)
= 360 - (A + B + C + D)/2
= 360 - 180
= 180

the sum of the opposite angles is 180 so it is a cyclic quadrilateral.
I hope it's helps u alot.

Thanku ....