A Visual Proof of Ptolemy's Theorem
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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 .[more]
Hover over , , , until you see a plus sign; then you can drag that vertex in its quadrant.[less]
Contributed by: Tomas Garza (April 2020)
Open content licensed under CC BY-NC-SA
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,
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
Finally, adding the two results,
 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.