Ptolemy's Theorem

ExampleAD⋅BC+AB⋅CD=AC⋅BD(6x3)+(7x4)=7AC18+28=7AC46=AC46/7=AC

Ptolemy's Theorem can be used to find the diagonals or the main lines of an inscribed quadrilateral, depending on what is given.Also used in Ptolemy's table of chords, for astronomy.

Let a convex quadrilateral ABCD be inscribed in a circle. Then the sum of the products of the two pairs of opposite sides equals the product of its two diagonals.AD⋅BC+AB⋅CD=AC⋅BD

What is it?

Example

On the diagonal BD locate a point M such that angles ACB and MCD be equal. Since angles BAC and BDC subtend the same arc, they are equal. Therefore, triangles ABC and DMC are similar. Thus we get CD/MD=AC/AB, or AB⋅CD=AC⋅MD.Now, angles BCM and ACD are also equal; so triangles BCM and ACD are similar which leads to BC/BM=AC/AD, or BC⋅AD=AC⋅BM. AB⋅CD+BC⋅AD=AC⋅MD+AC⋅BM=AC⋅BD

Proof

Ptolemy's TheoremBy Caleb Tussing

When to use it

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