# What is the origin of the "problem of Brahmagupta" of constructing inscribed quadrangle with given sides?

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.

• Inscribed where? Nov 13, 2021 at 20:35
• 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 Nov 13, 2021 at 20:43
• In general, Brahmagupta wrote two books (maybe more but these are available now), if there was a problem like that, it must be there. Nov 13, 2021 at 20:49
• 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? Nov 14, 2021 at 14:32
• @RayButterworth right Nov 14, 2021 at 18:39

"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..."