Miscellaneous triangle properties

Abstract

In this note we will discuss five topics from triangle geometry and occasionally encounter something new. Areal coordinates (presented as Appendix A in [1]) will be used throughout, with the exception of the final section where it is advantageous to employ Cartesians.The topics are:(1) examples of collinearities where the distance ratios for important points vary with triangle shape;(2) a pencil of lines which includes the Fermat, Napoleon and Vecten axes;(3) two more circles related to the Lester circle;(4) the family of rectangular hyperbolae which circumscribe the triangle of reference ABC and whose principal member is the Kiepert hyperbola [2];(5) Cartesian coordinates leading to a solution of the ‘Euler disc problem’ [3].