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]