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| − | == Statement ==
| + | #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.")
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| − | <center>[[Image:Stewart's_theorem.png]]</center>
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| − | == Proof ==
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| − | 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
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| − | *<math> n^{2} + d^{2} - 2\ce{nd}\cos{\angle CDA} = b^{2} </math>
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| − | *<math> m^{2} + d^{2} - 2\ce{md}\cos{\angle ADB} = c^{2} </math>
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| − | 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
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| − | *<math> \frac{n^2 + d^2 - b^2}{2\ce{nd}} = \cos{\angle CDA}</math>
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| − | *<math> \frac{c^2 - m^2 -d^2}{2\ce{md}} = \cos{\angle CDA}</math>
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| − | 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>.
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| − | However,
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| − | <math>m+n = a</math> so
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| − | *<math>m^2n + n^2m = (m + n)mn = amn</math> and
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| − | <cmath>d^2m + d^2n = d^2(m + n) = d^2a</cmath>
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| − | This simplifies our equation to yield <math>c^2n + b^2m = amn + d^2a,</math> or Stewart's Theorem.
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| − | == See also ==
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| − | * [[Menelaus' Theorem]]
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| − | * [[Ceva's Theorem]]
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| − | * [[Geometry]]
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| − | * [[Angle Bisector Theorem]]
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| − | [[Category:Geometry]]
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| − | [[Category:Theorems]]
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