Cramer–Castillon problem

Two solutions whose sides pass through

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]

The problem is as follows (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

  1. ^ Stark (2002), p. 1.
  2. ^ Wanner (2006), p. 59.
  3. ^ Ostermann & Wanner (2012), p. 176.

Bibliography

  • Dieudonné, Jean (1992). "Some problems in Classical Mathematics". Mathematics — The Music of Reason. Springer. pp. 77–101. doi:10.1007/978-3-662-35358-5_5. ISBN 978-3-642-08098-2.
  • Wanner, Gerhard (2006). "The Cramer–Castillon problem and Urquhart's 'most elementary´ theorem". Elemente der Mathematik. 61 (2): 58–64. doi:10.4171/EM/33. ISSN 0013-6018.
  • Stark, Maurice (2002). "Castillon's problem" (PDF). Archived from the original (PDF) on 2011-07-06.
  • 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.