Difference between revisions of "Stewart's Theorem"

Revision as of 10:53, 30 April 2008

Statement

Given a triangle $\triangle ABC$ with sides of length $a, b, c$ opposite vertices $A$ , $B$ , $C$ , respectively. If cevian $AD$ is drawn so that $BD = m$ , $DC = n$ and $AD = d$ , we have that $b^2m + c^2n = amn + d^2a$ . (This is also often written $cnc + bmb = man + dad$ , a form which invites mnemonic memorization, e.g. "A man and his dad put a bomb in the sink.")

Proof

Applying the Law of Cosines in triangle $\triangle ABD$ at angle $\angle ADB$ and in triangle $\triangle ACD$ at angle $\angle CDA$ , we get the equations

$n^{2} + d^{2} - 2nd\cos{\angle ADB} = c^{2}$
$m^{2} + d^{2} - 2md\cos{\angle CDA} = b^{2}$

Because angles $\angle ADB$ and $\angle CDA$ are supplementary, $m\angle ADB = 180^\circ - m\angle CDA$ . We can therefore solve both equations for the cosine term. Using the trigonometric identity $\cos{\theta} = -\cos{(180^\circ - \theta)}$ gives us

$\frac{n^2 + d^2 - b^2}{2nd} = \cos{\angle CDA}$
$\frac{c^2 - m^2 -d^2}{2md} = \cos{\angle CDA}$

Setting the two left-hand sides equal and clearing denominators, we arrive at the equation: $c^{2}n + b^{2}m=m^{2}n +n^{2}m + d^{2}m + d^{2}n$ . However, $m+n = a$ so $m^2n + n^2m = (m + n)mn$ and we can rewrite this as $c^2n + b^2m = amn + d^2a$ .

@@ Line 1: / Line 1: @@
 == Statement ==
-Given a [[triangle]] <math>\triangle ABC</math> with sides of length <math>a, b, c</math> opposite [[vertex | vertices]] <math>A</math>, <math>B</math>, <math>C</math>, respectively.  If [[cevian]] <math>AD</math> is drawn so that <math>BD = m</math>, <math>DC = n</math> and <math>AD = d</math>, we have that <math>b^2m + c^2n = amn + d^2a</math>.  (This is also often written <math> cnc + bmb = man + dad</math>, a form which invites mnemonic memorization.) A man and his dad put a bomb in the sink.
+Given a [[triangle]] <math>\triangle ABC</math> with sides of length <math>a, b, c</math> opposite [[vertex | vertices]] <math>A</math>, <math>B</math>, <math>C</math>, respectively.  If [[cevian]] <math>AD</math> is drawn so that <math>BD = m</math>, <math>DC = n</math> and <math>AD = d</math>, we have that <math>b^2m + c^2n = amn + d^2a</math>.  (This is also often written <math> cnc + bmb = man + dad</math>, a form which invites mnemonic memorization, e.g. "A man and his dad put a bomb in the sink.")
 <center>[[Image:Stewart's_theorem.png]]</center>

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Difference between revisions of "Stewart's Theorem"

Revision as of 10:53, 30 April 2008

Statement

Proof

See also