Cramer–Castillon problem

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Two solutions 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.

Bibliography