Examples for the application of a computer algebra system in geometry (Q1314955)

This short and elegant paper discusses the use of computer algebra systems for geometry. Recently systems have become available that cannot only do numerical calculations, but which can also manipulate algebraic expressions. Such packages are Mathematica and Maple, for example. Since these packages are also being used at schools, certain algebraic problems can also be treated, problems that would otherwise be exceedingly complex or require extensive calculations. Three examples from elementary geometry are presented using Mathematica. Firstly, a formula by Euler to compute the volume of a tetrahedron is verified. Secondly, an equation concerning a pair of isogonally conjugate points in a triangle is verified (two points \(F\) and \(G\) are isogonally conjugate in a triangle \(ABC\) if the segments \(AF\) and \(AG\) make the same angle with the bisector of \(A\) and they lie on opposite sides of it, and an analogous statement holds for \(B\) and \(C\)). Lastly, the computation of the so-called 3-fold angle point of a triangle is considered. It appears that this point lies on the Euler line, which can again be shown using the computer algebra system.