Stewart's Theorem
Revision as of 18:55, 23 June 2006 by Solafidefarms (talk | contribs) (Reverted edits by Solafidefarms (Solafidefarms); changed back to last version by Agolsme)
Contents
Statement
(awaiting image)
If a cevian of length t is drawn and divides side a into segments m and n, then
![$c^{2}n + b^{2}m = (m+n)(t^{2} + mn)$](http://latex.artofproblemsolving.com/5/8/4/584ba47a165beef167e4482a2f12ad9b242c296b.png)
Proof
For this proof we will use the law of cosines and the identity .
Label the triangle with a cevian extending from
onto
, label that point
. Let CA = n Let DB = m. Let AD = t. We can write two equations:
When we write everything in terms of cos(CDA) we have:
Now we set the two equal and arrive at Stewart's theorem:
Example
(awaiting addition)