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,

.

[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.