Details

1. The four vertices of a quadrilateral with vertices , , , all lie on the circumference of a circle.The lengths of the sides of are , , , and the lengths of its diagonals are and .

2. Draw the line (dashed red) so that , where is on the diagonal . That is, the two red angles are equal. Let and , so that .

3. The two blue angles are equal since they subtend the same blue arc from to . Hence, the two triangles and are similar, since they have two equal angles, the blue ones and the red angles plus .

4. Since the two shaded triangles are similar,

,

or .

5. The two orange angles are equal, , since they subtend the same arc . The triangle (with pink sides) and the triangle (with dark blue sides) are similar since the two angles and at vertex are equal from step 2. Then

,

or .

Finally, adding the two results,

.

Reference

[1] C. Alsina and R. B. Nelsen, When Less Is More: Visualizing Basic Inequalities, Washington, D.C.: Mathematical Association of America, 2009 p. 112. doi:10.5948/UPO9781614442028.