Wittgenstein, Germany

Sonnhmath@aol.com

In this contribution, I consider a problem or planar geometry that was inspired by another problem published in the German mathematics journal "Die Wurzel". The original problem, proposed by Prof. S. Arslanagic, consisted in characterizing by an equation those triangles which have their centre of gravity on the incircle. This led me to ask whether there exist triangles with the orthocentre and the circumcentre being situated on the periphery of the incircle. The problem is solved by the following steps: For a general triangle in a cartesian coordinate system, the coordinates of the orthocentre and the circumcentre are derived. By inserting these in the equation of the incircle, a system of polynomial equations with three variables a,b and c is obtained. This system is solved using the CAS Maple V and the package Groebner included therein. Result: Up to similarity, there is exactly one triangle with the orthocentre and the circumcentre situated on the periphery of the incircle.