Stewart's theorem

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Stewart's Theorem
Stewart's Theorem

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

a (p^2 + x y ) = b^2 x + c^2 y. \, or alternatively:
ap^2 = b^2 x + c^2 y - axy . \,

[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.

b^2 = p^2 + y^2 - 2 p y \cos { \theta } \,
c^2 = p^2 + x^2 + 2 p x \cos { \theta } \,

Multiply the first by x the latter by y :

x b^2 = x p^2 + x y^2 - 2 p x y \cos { \theta } \,
y c^2 = y p^2 + y x^2 + 2 p x y \cos { \theta } \,

Now add the two equations:

x b^2 + y c^2 = (x+y) p^2 + x y (x + y), \,

and this is Stewart's theorem.

[edit] See also

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