Still very Bretschneider, isn't it?

Given a general  convex quadrilateral with sides of lengths $a$, $b$, $c$, and $d$, the area is given by

$$\begin{align*} K&=\frac{1}{4}\sqrt{4p^2q^2-(b^2+d^2-a^2-b^2)^2}\tag{1}\\&=\sqrt{(s-a)(s-b)(s-c)(s-d)-\frac{1}{4}(ac+bd+pq)(ac+bd-pq)}\tag{2}\end{align*}$$

where $p$ and $q$ are the diagonal lengths and $s$ is the semiperimeter.

In MathWorld the American mathematician Julian Coolidge is credited with giving the second form of this formula, stating "here is one [formula] which, so far as I can find out, is new," while at the same time crediting Bretschneider and Strehlke with "rather clumsy" proofs of the related formula

$$\begin{align*}K&=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left(\frac{\alpha+\gamma}{2}\right)}\tag{3}\end{align*}$$

where $\alpha$ and $\gamma$ are two opposite angles of the quadrilateral.

The author of this note has realized that Coolidge's Formula is a easy consequence of Bretschneider's results (i.e., the Bretschneider's formula and Bretschneider's generalization of Ptolemy's Theorem).

In 1842 Bretschneider derived the following generalization of Ptolemy's theorem, regarding the product of the diagonals in a convex quadrilateral.

Theorem 1 (Bretschneider). Given a general  convex quadrilateral with sides of lengths $a$, $b$, $c$, and $d$, then

$$p^2q^2=a^2c^2+b^2d^2-2abcd\cos{(\alpha+\gamma})\tag{4}$$

where $p$ and $q$ are the diagonal lengths and $\alpha$ and $\gamma$ are two opposite angles of the quadrilateral.

We avoid proving Theorem 1; however, you can consult $[1]$ for that purpose. 

Using the cosine double angle formula and substituting in $(4)$,

$$\begin{align*}p^2q^2&=a^2c^2+b^2d^2-2abcd\left[2\cos^2{\left(\frac{\alpha+\gamma}{2}\right)}-1\right]\tag{5}\\&=(ac+bd)^2-4abcd\cos^2{\left(\frac{\alpha+\gamma}{2}\right)}\tag{6}\end{align*}$$

Substituting from Bretschneider's Formula, 

$$\begin{align*}p^2q^2&=(ac+bd)^2-4\left[K^2-(s-a)(s-b)(s-c)(s-d)\right]\tag{7}\end{align*}$$

Isolating $K$ and factorizing you get $(2)$. Still very Bretschneider, isn't it? Coolidge's proof can be found in $[2]$.

References

$[1]$ Andreescu, Titu & Andrica, Dorian, Complex Numbers from A to...Z, Birkhäuser, 2006, pp. 207–209.

$[2]$ Coolidge, J. L. "A Historically Interesting Formula for the Area of a Quadrilateral." Amer. Math. Monthly 46, 345-347, 1939.