Law of cosines, without the angle

IMG_20180203_091958.jpgI recently twice on the AMCs came across this problem solving technique for finding side lengths in a triangle with a cevian.  (A cevian is a segment from one vertex of a triangle to the opposite side.)  It uses the law of cosines, but the good news is it may not require an angle measure because it takes advantage of the fact that cos(180-\alpha)=-cos(\alpha).  Note in the diagram that the cevian creates these 2 angles.  By the law cosines a^2=c^2+e^2-2abcos(\alpha) and b^2=d^2+e^2-2abcos(180-\alpha) =  d^2+e^2+2abcos(\alpha).  Both equations have the length of the cevian, e, and the angle \alpha, so you can eliminate a variable and solve.

 

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