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 . ,

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 { heta } ,

:$$c^2 = p^2 + x^2 + 2 p x cos { heta }.,

Multiply the first by "x" the latter by "y" :

:$$x b^2 = x p^2 + x y^2 - 2 p x y cos { heta } ,

:$$y c^2 = y p^2 + y x^2 + 2 p x y cos { heta }.,

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.

ee also

*Apollonius' theorem

External links

* [http://planetmath.org/encyclopedia/StewartsTheorem.html Stewart's Theorem on PlanetMath]
* [http://planetmath.org/encyclopedia/ProofOfStewartsTheorem.html A proof of the theorem on PlanetMath]
* [http://mathworld.wolfram.com/StewartsTheorem.html Stewart's Theorem on MathWorld]
* [http://www.cut-the-knot.org/pythagoras/corollary.shtml#stewart Stewart's Theorem as a Corollary of the Pythagorean Theorem]