Law of cotangents

Statement

Using the usual notations for a triangle (see the figure at the upper right), where a, b, c are the lengths of the three sides, A, B, C are the vertices opposite those three respective sides, α, β, γ are the corresponding angles at those vertices, s is the semi-perimeter, that is, s = a + b + c/2, and r is the radius of the inscribed circle, the law of cotangents states that

cot ⁡ ( α 2 ) s − a = cot ⁡ ( β 2 ) s − b = cot ⁡ ( γ 2 ) s − c = 1 r

and furthermore that the inradius is given by

r = ( s − a ) ( s − b ) ( s − c ) s .

Proof

In the upper figure, the points of tangency of the incircle with the sides of the triangle break the perimeter into 6 segments, in 3 pairs. In each pair the segments are of equal length. For example, the 2 segments adjacent to vertex A are equal. If we pick one segment from each pair, their sum will be the semiperimeter s. An example of this is the segments shown in color in the figure. The two segments making up the red line add up to a, so the blue segment must be of length s − a. Obviously, the other five segments must also have lengths s − a, s − b, or s − c, as shown in the lower figure.

By inspection of the figure, using the definition of the cotangent function, we have

cot ⁡ ( α 2 ) = s − a r

and similarly for the other two angles, proving the first assertion.

For the second one—the inradius formula—we start from the general addition formula:

cot ⁡ ( u + v + w ) = cot ⁡ u + cot ⁡ v + cot ⁡ w − cot ⁡ u cot ⁡ v cot ⁡ w 1 − cot ⁡ u cot ⁡ v − cot ⁡ v cot ⁡ w − cot ⁡ w cot ⁡ u

Applying to cot(α/2 + β/2 + γ/2) = cot π/2 = 0, we obtain:

cot ⁡ ( α 2 ) cot ⁡ ( β 2 ) cot ⁡ ( γ 2 ) = cot ⁡ ( α 2 ) + cot ⁡ ( β 2 ) + cot ⁡ ( γ 2 )

(This is also the triple cotangent identity)

Substituting the values obtained in the first part, we get:

( s − a ) r ( s − b ) r ( s − c ) r = s − a r + s − b r + s − c r

But the right side equals s/r, and multiplying through by r³/s gives the value of r², proving the second assertion.

Some proofs using the law of cotangents

A number of other results can be derived from the law of cotangents.

Heron's formula. Note that the area of triangle ABC is also divided into 6 smaller triangles, also in 3 pairs, with the triangles in each pair having the same area. For example, the two triangles near vertex A, being right triangles of width s − a and height r, each have an area of 1/2r(s − a). So those two triangles together have an area of r(s − a), and the area S of the whole triangle is therefore

This gives the result S = √s(s − a)(s − b)(s − c) as required.

First Mollweide's formula. From the addition formula and the law of cotangents we have

This gives the result a − b c = sin ⁡ ( α 2 − β 2 ) cos ⁡ ( γ 2 ) as required.

Second Mollweide's formula. From the addition formula and the law of cotangents we have

Here, an extra step is required to transform a product into a sum, according to the sum/product formula. This gives the result b + a c = cos ⁡ ( α 2 − β 2 ) sin ⁡ ( γ 2 ) as required.

The law of tangents can also be derived from this (Silvester 2001, p. 99).