Newton's formulas relate the side lengths and opposite angles of a triangle. Let a triangle
have side lengths 
,
, and 
with opposite angles 
, 
, and 
. Then
 |  | ![(cos[1/2(B-C)])/(sin(1/2A))](/images/equations/NewtonsFormulas/Inline9.svg) |
(1)
|
 |  | ![(cos[1/2(C-A)])/(sin(1/2B))](/images/equations/NewtonsFormulas/Inline12.svg) |
(2)
|
 |  | ![(cos[1/2(A-B)])/(sin(1/2C)).](/images/equations/NewtonsFormulas/Inline15.svg) |
(3)
|
The companion difference identities are called Mollweide's formulas (Beyer 1987, p. 146), although the sum and difference identities are also grouped together as Mollweide's formula or Mollweide's equation (Karjanto 2011, pp. 70-71).
A generalization to a nondegenerate convex cyclic quadrilateral is given by García (2022, pp. 19-21). Let its successive
vertices be 
, 
, 
, and 
, with side lengths 
, 
, 
, and 
, and let its interior angles
be 
,
, 
, and 
, respectively. If its polygon
diagonals intersect at 
and 
, then
![(sin[1/2(alpha+beta)])/(cos[1/2(gamma-delta)])=(a+c)/(b+d)cot(1/2theta).](/images/equations/NewtonsFormulas/NumberedEquation1.svg) |
(4)
|
The triangle sum identity is recovered in the limit as two adjacent vertices coalesce on the circumcircle (García 2022, pp. 20-21).
