Difference between revisions of "Stewart's Theorem"
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== Proof == | == 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 | 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} - | + | *<math> n^{2} + d^{2} - 2\ce{nd}\cos{\angle CDA} = b^{2} </math> |
− | *<math> m^{2} + d^{2} - | + | *<math> m^{2} + d^{2} - 2\ce{md}\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 | 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}{ | + | *<math> \frac{n^2 + d^2 - b^2}{2\ce{nd}} = \cos{\angle CDA}</math> |
− | *<math> \frac{c^2 - m^2 -d^2}{ | + | *<math> \frac{c^2 - m^2 -d^2}{2\ce{md}} = \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>. | 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>. |
Revision as of 19:55, 1 December 2019
Statement
Given a triangle with sides of length
opposite vertices are
,
,
, respectively. If cevian
is drawn so that
,
and
, we have that
. (This is also often written
, a form which invites mnemonic memorization, i.e. "A man and his dad put a bomb in the sink.")
![Stewart's theorem.png](https://wiki-images.artofproblemsolving.com//b/b3/Stewart%27s_theorem.png)
Proof
Applying the Law of Cosines in triangle at angle
and in triangle
at angle
, we get the equations
Because angles and
are supplementary,
. We can therefore solve both equations for the cosine term. Using the trigonometric identity
gives us
Setting the two left-hand sides equal and clearing denominators, we arrive at the equation: .
However,
so
.