余切定理
![一个三角形。它的三个内角及其对边。](/Images/godic/202501/23/Triangle_with_notations_2.svg5648.png")
余切定理是三角学中关于三角形内切圆半径的定理。
假设α, β, 与γ是三角形的三个内角，a, b, 与c是与之对应的三个对边，若
-
![\zeta = \sqrt{\frac{1}{s} (s-a)(s-b)(s-c)}](/Images/godic/202501/23/0dd8678f65c7b5a764155546bee1b6275648.png")
（这个三角形的内切圆半径），其中： -
![s = \frac{a+b+c}{2 }](/Images/godic/202501/23/94e13c4f066fd8802476f8646c1a51345648.png")
（s就是三角形的半周长），
那幺余切定理告诉我们：
![\cot{ \frac{\alpha}{2 }} = \frac{s-a}{\zeta }](/Images/godic/202501/23/a73d9a451ea13524ebe013c0254aa9315648.png")
![\cot{ \frac{\beta}{2 }} = \frac{s-b}{\zeta }](/Images/godic/202501/23/9a95c4a98152c9cc0283ca4aac93ef8f5648.png")
![\cot{ \frac{\gamma}{2 }} = \frac{s-c}{\zeta }](/Images/godic/202501/23/49569cc4d461b2885d1b1e7ea3794fae5648.png")
还有
![\frac{\cot\frac{\alpha}{2}}{s-a} = \frac{\cot\frac{\beta}{2}}{s-b} = \frac{\cot\frac{\gamma}{2}}{s-c}.](/Images/godic/202501/23/84714f2e3ee06700452d36f19c22284c5648.png")
总而言之：余切定理就是:某个角一半的余切等于半周长减去这个角所对的边长再除以三角形的内切圆半径。
Law of cotangents
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英语百科
Law of cotangents 余切定理
![A triangle, showing the](/Images/godic/202501/23/Herontriangle1greek.svg5648.png")
![By the above reasoning, all six parts are as shown.](/Images/godic/202501/23/Herontriangle2greek.svg5648.png")
![](/Images/godic/202501/23/Webysther_20160305_-_Ciclo_trigonométrico.svg5648.png")
![Fig. 1. Notations usuelles dans un triangle.](/Images/godic/202501/23/Dreieck-mit-Inkreis25649.png")
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.
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.