Stewart's theorem

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In geometry, Stewart's theorem yields a relation between the lengths of the sides of a triangle and the length of segment from a vertex to a point on the opposite side.

Let a, b, c be the sides of a triangle. Let p be a segment from A to a point on a dividing a itself in x and y. Then

[edit] Proof

Call the point where a and p meet P. We start applying the law of cosines to the supplementary angles APB and APC.

Multiply the first by x the latter by y :

Now add the two equations:

and this is Stewart's theorem.

[edit] See also

• Apollonius' theorem

[edit] External links

• [1] on PlanetMath
• [2]: A proof of the theorem on PlanetMath
• [3] on Wolfram's MathWorld
• Stewart's Theorem as a Corollary of the Pythagorean Theorem