# 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.

Created with GeoGebra

### 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.)

- The Law of Cosines (Cosine Rule)
- The Illustrated Law of Cosines
- The Law of Sines and Cosines
- The Law of Cosines: Plane Tessellation
- The Law of Cosines: after Thâbit ibn Qurra
- The Law of Cosines: Unfolded Version
- The Law of Cosines (Independent of the Pythagorean Theorem)
- The Cosine Law by Similarity
- The Law of Cosines by Larry Hoehn
- The Law of Cosines - Another PWW
- The Law of Cosines - Yet Another PWW
- Law of Cosines by Ancient Sliding
- The Cosine Law: PWW by S. Kung

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