A Visual Proof of Ptolemy's Theorem

A significant result in classical geometry is Ptolemy's theorem: in a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides is equal to the product of the diagonals. This Demonstration presents a visual proof of the theorem, based on [1].
Hover over , , , until you see a plus sign; then you can drag that vertex in its quadrant.

THINGS TO TRY

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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.
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