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Difference between revisions of "Stewart's theorem"

m (→‎Proof)
m (→‎Proof)
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<cmath>m^2n + n^2m = (m + n)mn = amn</cmath> and  
 
<cmath>m^2n + n^2m = (m + n)mn = amn</cmath> and  
 
<cmath>d^2m + d^2n = d^2(m + n) = d^2a.</cmath>
 
<cmath>d^2m + d^2n = d^2(m + n) = d^2a.</cmath>
−
This simplifies our equation to yield <math>man + daf = bmb + cnc,</math> or Stewart's theorem.
+
This simplifies our equation to yield <math>man + dad = bmb + cnc,</math> or Stewart's theorem.
  
 
==Nearly Identical Video Proof with an Example by TheBeautyofMath==
 
==Nearly Identical Video Proof with an Example by TheBeautyofMath==

Revision as of 14:09, 28 December 2021

Statement

Given a triangle $\triangle ABC$ with sides of length $a, b, c$ opposite vertices are $A$, $B$, $C$, 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 $man + dad = bmb + cnc$, a form which invites mnemonic memorization, i.e. "A man and his dad put a bomb in the sink.")

Stewart's theorem.png

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

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

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 \[m^2n + n^2m = (m + n)mn = amn\] and \[d^2m + d^2n = d^2(m + n) = d^2a.\] This simplifies our equation to yield $man + dad = bmb + cnc,$ or Stewart's theorem.

Nearly Identical Video Proof with an Example by TheBeautyofMath

https://youtu.be/jEVMgWKQIW8

~IckMatrix

See also

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