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.

the law of cosines, acute case - PWW

the law of cosines, obtuse case - PWW

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


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