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I am looking for a source of the following construction problem:

Construct an inscribed quadrangle with given sides.

I know it under the name problem of Brahmagupta, but I do not know any evidence showing that Brahmagupta considered this problem (there is a relation to Brahmagupta's formula but very indirect).

P.S. Actually, I did not dig deep enuf --- sorry for that. It seems that the construction is given in Brahmasphutasiddhanta. At least Kim Plofker states so [12.39, Brahmasphutasiddhanta of Brahmagupta in Mathematics in India in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook p. 426]. In addition, Brahmagupta obtained the following formula for a diagonals [12.32 p. 425], say $x$, of inscribed quadrangle with sides $a$, $b$, $c$, and $d$: $$x=\sqrt{\tfrac{(a{\cdot}b+c{\cdot}d){\cdot}(a{\cdot}c+b{\cdot}d)}{a{\cdot}d+b{\cdot}c}},$$ and it leads to a solution of the construction problem as well.

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  • $\begingroup$ Inscribed where? $\endgroup$
    – markvs
    Commented Nov 13, 2021 at 20:35
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    $\begingroup$ You probably mean "inscribed in a circle". The guy seems to have many things named after him. For example, there is a Brahmagupta problem, but it is not what you want: en.wikipedia.org/wiki/Brahmagupta%27s_problem $\endgroup$
    – markvs
    Commented Nov 13, 2021 at 20:43
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    $\begingroup$ In general, Brahmagupta wrote two books (maybe more but these are available now), if there was a problem like that, it must be there. $\endgroup$
    – markvs
    Commented Nov 13, 2021 at 20:49
  • $\begingroup$ What exactly is the problem? My guess is that it is "given the lengths of 4 sides, construct a quadrilateral whose four corners are equal distance from a given point". Or is it something completely different? $\endgroup$
    – Ray Butterworth
    Commented Nov 14, 2021 at 14:32
  • $\begingroup$ @RayButterworth right $\endgroup$
    – Anton Petrunin
    Commented Nov 14, 2021 at 18:39

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Brahmagupta did not exactly consider this problem, he was rather interested in generalizing Heron's formula for the area of a triangle in terms of its sides to cyclic quadrilaterals, as they are now called. However, his surviving text was so obscure that after Colebrook's translation of Brahmasphuṭasiddhanta in 1817 European mathematicians were free to read all sorts of things into it. This particular reading is due to Chasles (Aperçu Historique sur l’Origine et le Développement des Méthodes en Géométrie, 1837), as is the problem of constructing cyclic quadrilaterals with rational sides. See Kichenassamy's Brahmagupta’s derivation of the area of a cyclic quadrilateral that gives a historical sketch and a modern reconstruction of Brahmagupta’s arguments:

"Chasles [1837, 420–447] sees in Brahmagupta’s Propositions XII.21–38 an outline of a general theory of quadrilaterals in which all the lines considered by Brahmagupta are rational. Indeed, XII.33–38 explain how to obtain, in terms of arbitrary quantities (rśi), the sides of distinguished quadrilaterals. Chasles considers that Brahmagupta’s propositions constitute steps leading up to the construction of such rational quadrilaterals... Chasles also points out that a given set of chords, $a,b,c$, and $d$ in a circle in general do not determine one but three essentially different cyclic quadrilaterals..."

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  • $\begingroup$ The problem should be known for a longer time, I would expect 1000 years at least. $\endgroup$
    – Anton Petrunin
    Commented Nov 14, 2021 at 18:41
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    $\begingroup$ @AntonPetrunin I'd be curious about a reference prior to Chasles. Nine point circle, "Napoleon's" theorem, Dandelin spheres, etc., "should" have been known to Greeks, but did not appear until the 19th century. $\endgroup$
    – Conifold
    Commented Nov 15, 2021 at 12:15
  • $\begingroup$ Actually, I found something (see the postscript in my question) --- maybe you have a comment on it? $\endgroup$
    – Anton Petrunin
    Commented Nov 15, 2021 at 17:52

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