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10 E.A. Jos´e Garc´ıa
Since the area of ABCD can be expressed as the sum of the areas of 4ABD and 4CBD,
which in turn can be written as ad sin α + bc sin γ , then we are done.
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It is interesting to note the resemblance of these area theorems to the identities (4), (1),
(5) and (6). Indeed, as Heron’s formula and Brahmagupta’s formula are both special cases of
Bretschneider’s formula, in the same way, the identities (4) and (1) are both special cases of
the identities (5) and (6). Actually, this better explains the development Heron-Brahmagupta-
Bretschneider. Finally we wonder how many other interesting implications the identities (5)
and (6) could have. What other research project could they inspire? For example, shall it be
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possible to obtain analogous identities in spherical or hyperbolic geometry? If so, how would
they relate to other well-known identities in such geometries? We leave the reader with these
intriguing questions in mind.
References
[1] R. C. Alperin, Heron’s Area Formula, The College Mathematics Journal 18(2) 137–138.
[2] G.A. Bajgonakova and A. Mednykh, On Bretschneider’s formula for a spherical quadrilateral,
Matematicheskie Zametki YAGU 1 (2012).
[3] G.A. Bajgonakova and A.Mednykh, On Bretschneider’s formula for a hyperbolic quadrilateral,
Matematicheskie Zametki YAGU 1 (2012).
[4] J. P. Ballantine, Note on Hero’s formula, Amer. Math. Monthly 61 (1945) 337.
[5] C. A. Bretschneider, Untersuchung der trigonometrischen Relationen des geradlinigen Viere-
ckes, Archiv der Mathl 2 (1842) 225–261.
[6] J. Casey, A Treatise On Plane Trigonometry, 1888, pp. 185-186.
[7] J. H. Conway and Peter Doyle, Personal email communication, 15 December 1997,
https://math.dartmouth.edu/ doyle/docs/heron/heron.txt
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[8] J. L. Coolidge, A Historically Interesting Formula for the Area of a Quadrilateral, Amer.
Math. Monthly 46 (1939) 345–347.
[9] E. J. Garc´ıa, Proofs and applications of two well-known formulae involving sine, cosine and
the semiperimeter of a triangle, from GeoDom, https://geometriadominicana.blogspot.
com/2020/06/another-proof-for-two-well-known.html
[10] A. Hess, A highway from Heron to Brahmagupta, Forum Geometricorum 12 (2012) 191–192.
[11] V. F. Ivanoﬀ, Solution to Problem E1376: Bretschneider’s Formula, Amer. Math. Monthly
67 (1960) 291–292.
[12] M. Josefsson, Calculations Concerning the Tangent Lengths and Tangency Chords of a
Tangential Quadrilate, Forum Geometricorum 10 (2010) 119–130.
[13] M. Josefsson, The area of a bicentric quadrilateral, Forum Geometricorum 11 (2011)
155–164.
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As suggested by work by G.A. Bajgonakova and A. Mednykh [2, 3].
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