# Cramer–Castillon problem

File:Cramer-Castillon Problem.png
To find the inscribed triangles in $Z$, whose sides pass through $A, B, C$

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

The problem consists of (see the image):

Given a circle $Z$ and three points $A, B, C$ in the same plane and not on $Z$, to construct every possible triangle inscribed in $Z$ whose sides (or their elongations) pass through $A, B, C$ 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 $n$ points.[3]

## References

1. Stark, page 1.
2. Wanner, page 59.
3. Ostermann and Wanner, page 176.