mg.metric geometry – 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

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

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

However, the author of this note has noticed that “Coolidge’s Formula” can be derived from two Bretschneider’s results (i.e, the Bretschneider’s formula and Bretschneider’s generalization of Ptolemy’s Theorem) . So if Bretschneider overlooked $(2)$, we can at least say that the latter is an easy consequence of his results.

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

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

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

Since the aim of this note is just to show that $(2)$ can be easily derived from Bretschneider’s results, we avoid proving Theorem 1. However, you can consult $(1)$ for that purpose.

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

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

Substituting from Bretschneider’s Formula,

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

Isolating $K$ and factorizing you get $(2)$. Still very Bretschneider, isn’t it?

So this is my concern: is this the proof given by Coolidge 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.