Difference between revisions of "Stewart's Theorem"

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== Statement ==
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#REDIRECT[[Stewart's theorem]]
Given a [[triangle]] <math>\triangle ABC</math> with sides of length <math>a, b, c</math> opposite [[vertex | vertices]] are <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>man + dad = bmb + cnc</math>, a form which invites mnemonic memorization, i.e. "A man and his dad put a bomb in the sink." When you're practicing to memorize this formula, never practice it in the library/airport or any other public place where other people can hear you.)
 
 
 
<center>[[Image:Stewart's_theorem.png]]</center>
 
 
 
== Proof ==
 
Applying the [[Law of Cosines]] in triangle <math>\triangle ABD</math> at [[angle]] <math>\angle ADB</math> and in triangle <math>\triangle ACD</math> at angle <math>\angle CDA</math>, we get the equations
 
*<math> n^{2} + d^{2} - 2nd\cos{\angle CDA} = b^{2} </math>
 
*<math> m^{2} + d^{2} - 2md\cos{\angle ADB} = c^{2} </math>
 
 
 
Because angles <math>\angle ADB</math> and <math>\angle CDA</math> are [[supplementary]], <math>m\angle ADB = 180^\circ - m\angle CDA</math>.  We can therefore solve both equations for the cosine term.  Using the [[trigonometric identity]] <math>\cos{\theta} = -\cos{(180^\circ - \theta)}</math> gives us
 
*<math> \frac{n^2 + d^2 - b^2}{2nd} = \cos{\angle CDA}</math>
 
*<math> \frac{c^2 - m^2 -d^2}{2md} = \cos{\angle CDA}</math>
 
 
 
Setting the two left-hand sides equal and clearing [[denominator]]s, we arrive at the equation: <math> c^{2}n + b^{2}m=m^{2}n +n^{2}m + d^{2}m + d^{2}n </math>.
 
However, <math>m+n = a</math> so <math>m^2n + n^2m = (m + n)mn</math> and we can rewrite this as <math>man + dad= bmb + cnc</math> (A man and his dad put a bomb in the sink). When you're practicing to memorize this formula, never practice it in the library or any other public place where other people can hear you.
 
 
 
== See also ==
 
* [[Menelaus' Theorem]]
 
* [[Ceva's Theorem]]
 
* [[Geometry]]
 
* [[Angle Bisector Theorem]]
 
 
 
[[Category:Geometry]]
 
 
 
[[Category:Theorems]]
 

Latest revision as of 16:22, 9 May 2021

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