Bretschneider's Formula

Theorem
Let $ABCD$ be a general quadrilateral.

Then the area $\mathcal A$ of $ABCD$ is given by:


 * $\mathcal A = \sqrt{\left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \left({s - d}\right) - a b c d \cos^2 \left({\dfrac {\alpha + \gamma} 2}\right)}$

where:
 * $a, b, c, d$ are the lengths of the sides of the quadrilateral
 * $s = \dfrac {a + b + c + d} 2$ is the semiperimeter
 * $\alpha$ and $\gamma$ are opposite angles.

Proof

 * Bretschneider's Formula.png

Let the area of $\triangle DAB$ and $\triangle BCD$ be $\mathcal A_1$ and $\mathcal A_2$.

From Area of Triangle in Terms of Two Sides and Angle:
 * $\mathcal A_1 = \dfrac {a b \sin \alpha} 2$ and $\mathcal A_2 = \dfrac {c d \sin \gamma} 2$

From to the second axiom of area, $\mathcal A = \mathcal A_1 + \mathcal A_2$, so:

The diagonal $p$ can be written in 2 ways using the Law of Cosines:
 * $p^2 = a^2 + b^2 - 2 a b \cos \alpha$
 * $p^2 = c^2 + d^2 - 2 c d \cos \gamma$

Equality is transitive, so:

Now add this equation to $(1)$. Then trigonometric identities can be used, as follows:

By expanding the square $\left({a^2 + b^2 - c^2 - d^2}\right)^2$:

Adding and subtracting $8 a b c d$ to and from the numerator of the first term of $(2)$:

allows the product $\left({-a + b + c + d}\right) \left({a - b + c + d}\right) \left({a + b - c + d}\right) \left({a + b + c - d}\right)$ to be formed:

Hence the result.

Also see

 * Brahmagupta's Formula is a specific version of Bretschneider's Formula for a cyclic quadrilateral.

In this case, from Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles, $\alpha + \gamma = 180^\circ$ and the formula becomes:
 * $\mathcal A = \sqrt{\left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \left({s - d}\right)}$


 * Heron's Formula is Brahmagupta's Formula for triangles, so $d = 0$ and the formula becomes:
 * $\mathcal A = \sqrt{s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right)}$

He published a proof in 1842.