# The Law of Cosines: What is this about?

A Mathematical Droodle

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

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Copyright © 1996-2018 Alexander BogomolnyThe applets suggests a proof of the Law of Cosines communicated to me by Douglas Rogers. The proof is a direct generalization of Thâbit ibn Qurra's proof of the Pythagorean proposition. The proof works for both obtuse and acute angles. The result is in accordance with II.12 and II.13, Euclid's statements for the two cases.

What if applet does not run? |

The construction is this: squares are formed outwardly on sides AC and BC of ΔABC and the diagram os complemented by three parallelograms - CTPR, STPM, and QRPN. The resulting heptagon ABQNPMS can be cut in two ways ...

The proof follows from an observation 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.

As Thâbit ibn Qurra's proof of the Pythagorean proposition that has an unfolded variant, so does its generalization above. The unfolded variant of the proof, appears on a separate page.

- 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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Copyright © 1996-2018 Alexander Bogomolny68676000