Page 16 - MATINF Nr. 13-14

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ARTICOLE SI NOTE DE MATEMATICA
,

A Generalization of the Law of Cotangents

e
Emmanuel Antonio Jos´ Garc´ıa 1

In trigonometry, the law of cotangents is a relationship among the side lengths of a triangle
0
0
and the cotangents of the halves of its angles [3, 4]. For a triangle with side lengths a , b , c 0
opposite the vertices A, B, C respectively, let
0
0
a + b + c 0
s = and r = inradius.
2
0
If the angles at A, B, C are α , β, γ, then
0
cot(α /2) cot(β/2) cot(γ/2) 1
= = = .
s − a 0 s − b 0 s − c 0 r

In this note, we generalize the law of cotangents to cyclic quadrilaterals. Let ABCD be a
cyclic quadrilateral with side lengths

a + b + c + d
|AB| = a, |BC| = b, |CD| = c, |DA| = d, s = .
2
Set α = ∠BAD, β = ∠ABC, γ = ∠BCD, δ = ∠CDA (see Figure 1).

Figure 1. A cyclic quadrilateral ABCD.

Let ∆ denote the area of ABCD.
1
Professor, CIDIC-Universidad UTE, Santo Domingo, Dominican Republic, emmanuelgeogarcia@gmail.com

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