Cramer–Castillon problem

In geometry, the Cramer–Castillon problem is a problem stated by the Genevan mathematician Gabriel Cramer solved by the Italian mathematician, resident in Berlin, Jean de Castillon in 1776.^[1]

Two solutions whose sides pass through ${\displaystyle A,B,C}$

The problem is as follows (see the image): given a circle ${\displaystyle Z}$ and three points ${\displaystyle A,B,C}$ in the same plane and not on ${\displaystyle Z}$, to construct every possible triangle inscribed in ${\displaystyle Z}$ whose sides (or their elongations) pass through ${\displaystyle A,B,C}$ respectively.

Centuries before, Pappus of Alexandria had solved a special case: when the three points are collinear. But the general case had the reputation of being very difficult.^[2] After the geometrical construction of Castillon, Lagrange found an analytic solution, easier than Castillon's. In the beginning of the 19th century, Lazare Carnot generalized it to ${\displaystyle n}$ points.^[3]