Difference between revisions of "Stewart's theorem"
(→Proof) |
(→Proof 2 (Pythagorean Theorem)) |
||
Line 45: | Line 45: | ||
<cmath>man+dad = bmb+cnc</cmath> | <cmath>man+dad = bmb+cnc</cmath> | ||
+ | |||
+ | ~sml1809 | ||
==Nearly Identical Video Proof with an Example by TheBeautyofMath== | ==Nearly Identical Video Proof with an Example by TheBeautyofMath== |
Revision as of 02:58, 10 March 2022
Contents
[hide]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.")
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.
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
Nearly Identical Video Proof with an Example by TheBeautyofMath
~IceMatrix