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A generalization of Mollweide’s formula, rather Newton’s

Emmanuel Antonio Jos´ Garc´ıa 1
e

Mollweide’s formula, sometimes also referred to as Mollweide’s equation, is a set of two
relationships between sides and angles in a triangle. This equation is particularly useful in
checking one’s result after solving an oblique triangle since all six components of the triangle
are involved.

Let a = |BC|, b = |AC|, and c = |AB| be the lengths of the three sides of a triangle. Let
α, β, and γ be the measures of the angles opposite those three sides respectively. Mollweide’s
formula states that
1
a + b cos (α − β)
2
= (1)
1
c sin γ
2
and
1
a − b sin (α − β)
= 2 . (2)
1
c cos γ
2
The equations adopt their name from a German mathematician and astronomer Karl Brandan
Mollweide. Nonetheless, this pair of equations was discovered earlier by Isaac Newton. In
fact, formula (1) is also known as Newton’s formula. An excellent overview of the history of
Mollweide’s formula is given by Wu [5]. For proofs of the Mollweide’s formula we invite the
readers to see Karjanto’s article on the subject [4].
The following theorem generalizes formula (1). We recall that a cyclic quadrilateral is a
quadrilateral whose vertices all lie on a single circle.

Theorem 1. Let a = |AB|, b = |BC|, c = |CD| and d = |AD| be the sides of a cyclic convex
quadrilateral. Let ∠DAB = α, ∠ABC = β, ∠BCD = γ and ∠CDA = δ. If AC and BD
intersect at E, denote ∠CED = θ. Then the following identity holds
1
sin (α + β) a + c 1
2 = cot θ. (3)
1
cos (γ − δ) b + d 2
2
See Figure 1 for an example of the situation described in Theorem 1.

Proof. We take advantage of the cyclic nature of the half-angle formulas [2, p. 186] in combination
with the formulas of compound angles.
1
1
1
1
1
sin (α + β) sin α cos β + cos α sin β
2 = 2 2 2 2 .
1
1
1
1
1
cos (γ − δ) cos γ cos δ + sin γ sin δ
2 2 2 2 2
Substituting from the half-angle formulas
» (s−a)(s−d) » (s−c)(s−d) » (s−b)(s−c) » (s−a)(s−b)
1
sin (α + β) +
2 ad+bc ab+cd ad+bc ab+cd ,
1
cos (γ − δ) = » (s−a)(s−d) » (s−a)(s−b) » (s−b)(s−c) » (s−c)(s−d)
2 +
ad+bc ab+cd ad+bc ab+cd
1
Professor, CIDIC-Universidad UTE, Santo Domingo, Dominican Republic, emmanuelgeogarcia@gmail.com
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