Renowned British mathematician John Conway, in correspondence with Peter Doyle, used two trigonometric formulae to prove Heron's formula. John Casey, in his book "A Treatise of Plane Trigonometry" used two other trigonometric formulae to prove the Brahmagupta's formula. It turns out that the two trigonometric formulae used by Casey for a cyclic quadrilateral generalize the two used by Conway for a triangle. No one seems to have wondered if the two formulae used by Casey could be generalized to a general quadrilateral and use it to prove the Bretschneider's formula. The last link in this chain is precisely what my last article is about!
Article: Two Identities and their Consequences
Abstract. In this note we prove the Heron’s formula (although known, see Conway’s dicussion in $[7]$), the Brahmagupta’s formula (also known, see $[6]$) and the formula for the area of a bicentric quadrilateral (possibly new, see $[12, 13]$), $\sqrt{abcd}$, based on two lesser-known trigonometric formulae $[6, 16]$ involving sine, cosine, the semiperimeter and the side lenghts of a cyclic quadrilateral. Once the two trigonometric formulae have been established (and the necessary adjustments made), the proofs of these area theorems are greatly simplified. Furthermore, we present a generalization of the two aforementioned trigonometric formulae and use it to give an alternative proof of Bretschneider’s formula. Since all these area theorems can be derived from this new generalization, the approach presented in this note, unlike others, provides a more holistic view of these theorems. Our main result for a general convex quadrilateral are the identities
\[ad\sin^2{\frac{\alpha}{2}}+bc\cos^2{\frac{\gamma}{2}}=(s-a)(s-d)\]
and
\[bc\sin^2{\frac{\gamma}{2}}+ad\cos^2{\frac{\alpha}{2}}=(s-b)(s-c),\]
where $a$, $b$, $c$, $d$ are the sides lengths, $s$ is the semiperimeter, and $\alpha$ and $\gamma$ are opposite angles.
Regarding the novelty of the results and proofs presented in this article, I have consulted Martin Josefsson (whom I consider an expert on these issues) and this was the message he sent me:
"Dear Emmanuel,
I like your paper, especially how you put these important formulas in a single framwork. I cannot say that I remember seeing the identities (4) and (5) anywhere else before.
I have seen (somewhere on the Internet) that proof of Heron's formula using half angle triangle formulas before - but not from your point of view of using cyclic quadrilateral formulas and setting one side = 0. These cyclic quadrilateral half angle formulas are, as you say, not so well known, but both them and your proof of Brahmagupta's formula can be found - more or less in the same way, but with fewer details - in Casey's 1888 book "A Treatise on Plane Trigonometry", see the attachment.
About the structure of the article I have not much to say, except perhaps to eliminate the penultimate step in the proofs of Theorems 1 and 5 where you explain how to introduce the semiperimeter in the formulas.
Even though much has already been written about these formulas, the ideas for proving Bretschneider' formula and the area of a bicentric quadrilateral are novel as far as I know. I hope you get your paper published.
Best regards,
Martin"
The identities $(4)$ and $(5)$ mentioned by Martin are now identities $(5)$ and $(6)$ in the present version of the paper. Martin's work on these issues can be found in the following link:
Below I've added an incomplete concept map of identities $(5)$ and $(6)$ so you can better appreciate the way they relate to other well-known identities.
