Law of cotangents

A triangle, showing the "incircle" and the partitioning of the sides. The angle bisectors meet at the incenter, which is the center of the incircle.

By the above reasoning, all six parts are as shown.

In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles.^[1]^[2]

Just as three quantities whose equality is expressed by the law of sines are equal to the diameter of the circumscribed circle of the triangle (or to its reciprocal, depending on how the law is expressed), so also the law of cotangents relates the radius of the inscribed circle of a triangle (the inradius) to its sides and angles.

^ The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.
^ It is called the 'theorem of the cotangents' in Apolinar, Efraín (2023). Illustrated glossary for school mathematics. pp. 260–261. ISBN 9786072941311.

[1] The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.

[2] It is called the 'theorem of the cotangents' in Apolinar, Efraín (2023). Illustrated glossary for school mathematics. pp. 260–261. ISBN 9786072941311.

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