Difference between revisions of "Stewart's theorem"
Jayevvms123 (talk | contribs) m (→Statement) |
|||
Line 50: | Line 50: | ||
~sml1809 | ~sml1809 | ||
+ | ==Proof 3 (Barycentrics)== | ||
+ | Let the following points have the following coordinates: | ||
+ | |||
+ | <math>A: (1,0,0)</math> | ||
+ | |||
+ | <math>B: (0,1,0)</math> | ||
+ | |||
+ | <math>C: (0,0,1)</math> | ||
+ | |||
+ | <math>D: (0, \frac{n}{m+n},\frac{m}{m+n})</math> | ||
+ | |||
+ | Our displacement vector <math>\overrightarrow{AD}</math> has coordinates <math>(1, -\frac{n}{m+n}, -\frac{m}{m+n})</math>. Plugging this into the barycentric distance formula, we obtain <cmath>d^2=-(m+n)^2 \left(\frac{mn}{(m+n)^2} \right)-b^2 \left ( \frac{-m}{m+n} \right)-c^2 \left(\frac{-n}{m+n}\right)=-mn+\frac{b^2m+c^2n}{m+n}</cmath> Multiplying by <math>m+n</math>, we get <math>d^2(m+n)+mn(m+n)=b^2m+c^2n</math>. Substituting <math>m+n</math> with <math>a</math>, we find Stewart's Theorem: <cmath>\boxed{d^2a+amn=b^2m+c^2n}</cmath> | ||
+ | |||
+ | ~kn07 | ||
==Nearly Identical Video Proof with an Example by TheBeautyofMath== | ==Nearly Identical Video Proof with an Example by TheBeautyofMath== | ||
https://youtu.be/jEVMgWKQIW8 | https://youtu.be/jEVMgWKQIW8 |
Revision as of 19:11, 27 January 2023
Contents
[hide]Statement
Given a triangle with sides of length and opposite vertices , , , respectively. If cevian is drawn so that , and , we have that . (This is also often written , a phrase which invites mnemonic memorization, i.e. "A man and his dad put a bomb in the sink.") That is Stewart's Theorem. I know, it's easy to memorize.
Proof 1
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 and This simplifies our equation to yield or Stewart's theorem.
Good Job! You mastered Stewart's Theorem.
Proof 2 (Pythagorean Theorem)
Let the altitude from to meet at . Let , , and . So, applying Pythagorean Theorem on yields
Since ,
Applying Pythagorean on yields
Substituting , , and in and gives
Notice that
are equal to each other. Thus, Rearranging the equation gives Stewart's Theorem:
~sml1809
Proof 3 (Barycentrics)
Let the following points have the following coordinates:
Our displacement vector has coordinates . Plugging this into the barycentric distance formula, we obtain Multiplying by , we get . Substituting with , we find Stewart's Theorem:
~kn07
Nearly Identical Video Proof with an Example by TheBeautyofMath
~IceMatrix