C-orthocenter and C-orthocentric systems in Minkowski planes

TL;DR

This paper extends classical triangle geometry concepts like the orthocenter and Euler points to Minkowski planes, exploring properties, relationships with Feuerbach's circle, and invariance under homothety.

Contribution

It introduces the notion of C-orthocenter in Minkowski planes and studies C-orthocentric systems, establishing their properties and relations with orthogonality and Feuerbach's circle.

Findings

Properties of C-orthocenter in Minkowski planes are characterized.

C-orthocentric systems are shown to be invariant under homothety.

Relations between C-orthocentric systems and classical geometric notions are established.

Abstract

Usando la noci\'on de C-ortocentro se extienden, a planos de Minkowski en general, nociones de la geometr\'ia cl\'asica relacionadas con un tri\'angulo, como por ejemplo: puntos de Euler, tri\'angulo de Euler, puntos de Poncelet. Se muestran propiedades de estas nociones y sus relaciones con la circunferencia de Feuerbach. Se estudian sistemas C-ortoc\'entricos formados por puntos presentes en dichas nociones y se establecen relaciones con la ortogonalidad is\'osceles y cordal. Adem\'as, se prueba que la imagen homot\'etica de un sistema C-ortoc\'entrico es un sistema C-ortoc\'entrico. -- Using the notion of C-orthocenter, notions of the classic euclidean geometry related with a triangle, as for example: Euler points; Euler's triangle; and Poncelet's points, are extended to Minkowski planes in general. Properties of these notions and their relations with the Feuerbach's circle, are shown. C-orthocentric systems formed by points in the above notions are studied and relations with the isosceles and chordal orthogonality, are established. In addition, there is proved that the homothetic image of a C-orthocentric system is a C-orthocentric system.