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[[File:Herontriangle1.svg|thumb|270px|right||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 and excircles of a triangle|incircle]].]]
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[[File:Herontriangle2.svg|thumb|270px|right||By the above reasoning, all 6 parts are as shown.]]
{{Trigonometry}}
In [[trigonometry]], the '''law of cotangents'''<ref>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.</ref> relates the radius of the [[inscribed circle]] of a [[triangle]] (the [[inradius]]) to its sides and angles.
 
==Statement==
Using the usual notations for a triangle (see the figure at the upper right), where <math>a, b, c</math> are the lengths of the three sides, <math>A, B, C</math> are the angles opposite those three respective sides, <math>s</math> is the semi-perimeter, that is, <math>s = (a+b+c)/2</math>, and <math>r</math> is the radius of the inscribed circle, the law of [[trigonometric function|cotangent]]s states that:
::<math>\frac{\cot(A/2)}{s-a} = \frac{\cot(B/2)}{s-b} = \frac{\cot(C/2)}{s-c} = \frac{1}{r}\,</math>.
 
And furthermore that the inradius is given by:
::<math>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}\,</math>
 
==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&minus;a''. Obviously, the other 5 segments must also have lengths ''s&minus;a'', ''s&minus;b'', or ''s&minus;c'', as shown in the lower figure.
 
By inspection of the figure, using the definition of the cotangent function, we have:
::<math>\cot(A/2) =\frac{s-a}{r}\,</math>.
and similarly for the other two.
 
To get the inradius, note that the area of the triangle 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&minus;a'' and height ''r'', each have an area of <math>\tfrac{1}{2}r(s-a)</math>. So those two triangles together have an area of <math>r(s-a)\,</math>, and the area of the whole triangle is therefore:
::<math>
\begin{align}
\text{Area }&= r(s-a) + r(s-b) + r(s-c) = r(s-a + s-b + s-c) \\[8pt]
&= r(3s - (a+b+c)) = r(3s - 2s) = rs \\[8pt]
\end{align}
</math>
 
But, by [[Heron's formula]], the area is also
::<math>\sqrt{s(s-a)(s-b)(s-c)}\,</math>
so
::<math>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}\,</math>
 
==Some proofs using the Law of cotangents==
A number of other results can be derived from the law of cotangents.
 
*First [[Mollweide's formula]]. From the addition formula and the Law of cotangents we have:
:<math>{\frac {\sin \left( \alpha/2-\beta/2 \right) }{\sin \left( \alpha/2+\beta/2 \right) }}= {\frac {\cot \left( \beta/2 \right) -\cot \left( \alpha/2 \right) }{\cot \left( \beta/2 \right) +\cot \left( \alpha/2 \right) }}=  {\frac {a-b}{2s-a-b}}</math>.
 
:This gives the result: <math>{\dfrac {a-b}{c}}={\dfrac {\sin \left( \alpha/2-\beta/2 \right)}{\cos \left( \gamma/2 \right) }}</math> as required.
 
*Second [[Mollweide's formula]]. From the addition formula and the Law of cotangents we have:
:<math>{\frac {\cos \left( \alpha/2-\beta/2 \right) }{\cos \left( \alpha/2+\beta/2 \right) }} =
{\frac {\cot \left( \alpha/2 \right) \cot \left( \beta/2 \right) +1}{\cot \left( \alpha/2 \right) \cot \left( \beta/2 \right) -1}} =
{\frac {\cot \left( \alpha/2 \right) +\cot \left( \beta/2 \right) +2\,\cot \left( \gamma/2 \right) }{\cot \left( \alpha/2 \right) +\cot \left( \beta/2 \right) }} =
{\frac {4\,s-a-b-2\,c}{2\,s-a-b}}</math>.
:Here, an extra step is required to transform a product into a sum, according to the sum/product formula.  
:This gives the result: <math>\dfrac {b+a}{c} = \dfrac{\cos \left( \alpha/2-\beta/2 \right) }{\sin \left( \gamma/2 \right) }</math> as required.
 
*The [[Law of tangents]] can also be derived from this {{harv|Silvester|2001|p=99}}, as can the [[Pythagorean theorem]].
 
==See also==
*[[Law of sines]]
*[[Law of cosines]]
*[[Law of tangents]]
*[[Mollweide's formula]]
*{{cite web|url=http://formulasheet.com/db/science/mathematics/trigonometric_laws_and_theorems/law_of_cotangents|title=Formula sheet database - law of cotangents|accessdate=October 30, 2013}}
 
==References==
{{reflist}}
* {{cite book
|title=Geometry: Ancient and Modern
|last=Silvester|first=John R.  
|publisher=Oxford University Press
|year=2001
|isbn=9780198508250
|pages=313
|ref=harv
}}
 
[[Category:Trigonometry]]

Latest revision as of 08:53, 14 August 2014

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