Cramer–Castillon problem

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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]


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


External links

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