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In trigonometry, the law of cotangents relates the radius of the inscribed circle of a triangle to its sides and angles.

When a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of cotangents states that if

\( \zeta = \sqrt{\frac{1}{s} (s-a)(s-b)(s-c)} \) (the radius of the inscribed circle for the triangle) and
s = \frac{a+b+c}{2 } (the semiperimeter for the triangle),

then the following all form the law of cotangents:[1]

\( \cot{ \frac{\alpha}{2 }} = \frac{s-a}{\zeta } \)

\( \cot{ \frac{\beta}{2 }} = \frac{s-b}{\zeta } \)

\( \cot{ \frac{\gamma}{2 }} = \frac{s-c}{\zeta } \)

It follows that

\( \frac{\cot(\alpha/2)}{s-a} = \frac{\cot(\beta/2)}{s-b} = \frac{\cot(\gamma/2)}{s-c}. \)

In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.
See also

Law of sines
Law of cosines
Law of tangents
Mollweide's formula
Tangent half-angle formula

References

^ 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.

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