Bretschneider's formula

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Bretschneider's formula gives the area of any quadrilateral. If the sides in order are a, b, c, d and the semiperimeter is s = (a + b + c + d)/2, then with A and C being two opposite angles, the area K is

K = sqrt( (s − a)(s − b)(s − c)(s − d) − abcd cos^2((A + C)/2) ).

The formula works for convex, concave, and crossed quadrilaterals (using directed angles so A + B + C + D equals 360° or 720° as appropriate). If the quadrilateral is cyclic (A + C = 180°), the cosine term disappears and you get Brahmagupta’s formula: K = sqrt( (s − a)(s − b)(s − c)(s − d) ). Bretschneider’s formula thus generalizes both Brahmagupta’s and Heron’s formulas.