Cramer–Castillon problem

To find the inscribed triangles in $Z$, whose sides pass through $A,B,C$

In geometry, the Cramer-Castillon's Problem is a problem stated by the swiss mathematician Gabriel Cramer solved by the italian mathematician, resident in Berlin, Jean de Castillon in 1776.

The problem consists on (see the image):

Let be a circle $Z$ and three points $A,B,C$ in the same plane and not in $Z$, to construct every posible triangles inscribed in $Z$, whose sides (or his elongations) pass through $A,B,C$ respectively.

Centurys before, Pappus of Alexandria has been solved an special case: when the three points ara aligned. But the general case, had the reputation of being very difficult.

After the geometrical construction of Castillon, Lagrange found an analytic solution, easier than Castillon's. In the begining of 19th century, Lazare Carnot generalized it to $n$ points.

Bibliography

• Dieudonné, Jean (1992). "Some problems in Classical Mathematics". Mathematics — The Music of Reason. Springer. pp. 77–101. ISBN 978-3-642-08098-2. {{cite book}}: Cite has empty unknown parameters: |data=, |lloc=, |citació=, |volum=, |pàgina=, |col·lecció=, |edició=, |mes=, and |consulta= (help)
• 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. {{cite book}}: Cite has empty unknown parameters: |lloc=, |citació=, |volum=, |pàgina=, |urlchapter=, |col·lecció=, |edició=, |mes=, |data=, and |consulta= (help)
• Template:Cite article

External links

• Stark, Maurice (2002). "Castillon's problem" (PDF).