The Law of Cosines: What is this about?
A Mathematical Droodle


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

(The three vertices of triangle ABC are draggable as is the triangle itself.)

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

The applets suggests a proof of the Law of Cosines. The proof is a modification of the one communicated to me by Douglas Rogers. The latter is a direct generalization of Thâbit ibn Qurra's proof of the Pythagorean proposition. The latter has an unfolded version, which is generalized by the present proof.

A proof without words: cosine law by dissection

As in its relative, an important observation is that CP is equal and perpendicular to AB. Indeed, triangles ABC and CTR are friendly, which implies that the median of the latter through the vertex C is altitude of the former. But, since CTPR is a parallelogram, its diagonals divide each other into equal halves, CP is indeed the median of ΔCTR.

Also, the whole diagram appears to be symmetric in the midpoint O of AB.

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Copyright © 1996-2017 Alexander Bogomolny

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