The angle bisectors of a quadrilateral form a cyclic quadrilateral
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In the picture above, we are given a quadrilateral \(ABCD\). The green lines are the angle bisectors of each of the internal angles of the quadrilateral. These angle bisectors intersect at the points \(W\), \(X\), \(Y\) and \(Z\). We proved in Lecture 8 that the quadrilateral \(WXYZ\) is cyclic, which you can see in the above picture as the orange circle is the circumcircle of the quadrilateral \(WXYZ\).
The above picture also shows the proof. The angles are coloured in four different colours. All the angles of the same colour add up to \(180^\circ\), since they form the angles in a triangle. Moreover, the sum of all the internal angles in a quadrilateral is \(360^\circ\) and so the sum of all the angle bisectors is \(180^\circ\). Therefore, in the quadrilateral \(WXYZ\), the opposite angles must sum to \(180^\circ\), which is exactly the required condition for \(WXYZ\) to be cyclic.
