# Difference between revisions of "Stewart's Theorem"

m (→Statement) |
(→Proof) |
||

Line 14: | Line 14: | ||

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>. | ||

− | However, <math>m+n = a</math> so <math>m^2n + n^2m = (m + n)mn</math> and we can rewrite this as <math>man + dad= bmb + cnc</math> (A man and his dad put a bomb in the sink). When you're practicing to memorize this formula, never practice it in the library or any other public place where other people can hear you. | + | However, <math>m+n = a</math> so <math>m^2n + n^2m = (m + n)mn</math> and we can rewrite this as <math>man + dad= bmb + cnc</math> (A man and his dad put a bomb in the sink). When you're practicing to memorize this formula, never practice it in the library or any other public place where other people can hear you. LOL! |

== See also == | == See also == |

## Revision as of 16:45, 9 March 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." When you're practicing to memorize this formula, never practice it in the library/airport or any other public place where other people can hear you.)

## 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 and we can rewrite this as (A man and his dad put a bomb in the sink). When you're practicing to memorize this formula, never practice it in the library or any other public place where other people can hear you. LOL!