Let

be a cyclic quadrilateral and

be the quadrilateral formed by the angle bisectors of angle

Required to prove:

is a cyclic quadrilateral.

Proof:

By angle sum property of a triangle

In

And, in

Adding (i) and (ii), we get

But,

Therefore,

[From (iii)]

But, these are the sum of opposite angles of quadrilateral

Therefore,

Quadrilateral

is also a cyclic quadrilateral.