Conway's dream (Q6169821)

Brahmagupta's formula calculates the area of a cyclic quadrilateral from the length of its edges. A quite similar formula for triangles is known as Heron's formula, and in fact Heron's formula can be regarded as a limiting case of Brahmagupta's formula. Since long a pure geometric proof for Heron's formula is known, but not for the formula of Brahmagupta. To find such a proof was a dream of John Conway -- and now Sam Vandervelde presents such a proof. The author starts with the proof of Heron's formula. This formula relies on the area of the triangle regarded as function of the semiperimeter, the inradius and an exradius. To get rid of the exradius, the author needs the similarity of two triangles that involve the common edge. For the cyclic quadrilateral, the author uses a similar approach. He extends its edges and add the circles externally tangent to them and to two neighbouring extensions. Then the centres and the feet of the circles at opposite edges define a polygon that embeddes the given cyclic quadrilateral. A short calculation gives the area of the quadrilateral as a function of the edges and the exradii. Again the author uses similar triangles involving the centres of the excircles to get rid of the exradii. The similarity of the triangles depends on the fact that two opposite angles of the quadrilateral sum up to \(\pi\) if it is cyclic. Finally, some calculations transform the product of egdes and exradii into Bramagupta's formula.