# Cramer–Castillon problem

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 and three points in the same plane and not on , to construct every possible triangle inscribed in whose sides (or their elongations) pass through 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 points.^{[3]}

## References

## Bibliography

- Dieudonné, Jean (1992). "Some problems in Classical Mathematics".
*Mathematics — The Music of Reason*. Springer. pp. 77–101. ISBN 978-3-642-08098-2.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Ostermann, Alexander; Wanner, Gerhard (2012). "6.9 The Cramer–Castillon problem".
*Geometry by Its History*. Springer. pp. 175–178. ISBN 978-3-642-29162-3.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Wanner, Gerhard (2006). "The Cramer–Castillon problem and Urquhart's `most elementary´ theorem".
*Elemente der Mathematik*. Vol. 61 (Num. 2). pp. 58–64. doi:10.4171/EM/33. ISSN 0013-6018.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

## External links

- Stark, Maurice (2002). "Castillon's problem" (PDF).<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>