# The Law of Cosines - Another PWW

The applet below illustrates a proof without words of the *Law of Cosines* that establishes a relationship between the angles and the side lengths of \(\Delta ABC\):

\(c^{2} = a^{2} + b^{2} - 2ab\cdot \mbox{cos}\gamma,\)

where \(\gamma\) is the angle in \(\Delta ABC\) opposite side \(c\).

There are four draggable points: the three vertices of the reference triangle and one on the resulting trapezoid.

### Proof

The applet illustrates the cases of acute and abtuse triangles, making it it clear that in irder to obtain \(c^2\), for acute triangles, \(2ab\space\mbox{cos}\gamma\) needs to be subtracted from \(a^{2}+b^{2}\), while for obtuse triangles it needs to be added.

(There are several theorems that are proved by similar technique.)

|Contact| |Front page| |Content| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny