The quadrilateral ABCD is inscribed in the circle K – the point of intersection of the diagonals AC

The quadrilateral ABCD is inscribed in the circle K – the point of intersection of the diagonals AC and BD, show that AK * CK = BK * DK

This quadrilateral, inscribed in a circle, is divided by its diagonals into two pairwise similar triangles.

Triangles AKD and BKC, AKB and CKD are similar, since in each pair of triangles all three angles are respectively equal to each other.

Consider triangles AKD and BKD. <CAD = <CBD, since they rely on a common CD arc. <ADB = <ACB, as they are based on a common arc AB. <AKD = <BKC, as vertical angles.

Let us write the ratios for the sides in similar triangles: sides against equal angles are proportional. Then:

KD / KC = AK / BK, AK * CK = BK * DK.