2025/11/17 by Kazimierz Chomicz, Chomicz, Kazimierz, Miłosz Płatek +5
Mathematics · #51M05 #51M15 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Finite Group Theory Research #Mathematics and Applications #Metric Geometry (math.MG) #Primary 51M04
paper · pdf · doi:10.48550/arxiv.2511.13298
openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
We consider the following configuration. Let ABCD be a cyclic quadrilateral with circumcenter O, and for each vertex X, let HX be the orthocenter of the triangle formed by the other three. Then A, B, C, D, HA, HB, HC, HD all lie on a single conic. In this paper we study a certain generalization of this fact as follows. For an arbitrary point PD on the Euler line of \triangle ABC, we define corresponding points PA, PB, PC on the respective Euler lines such that the ratio PXHX : PXO is constant for all X. We show that the four vertices A,B,C,D and the four isogonal conjugates QA, QB ,QC ,QD of the points PX all lie on a single conic. This result is given distinct treatments, synthetic, projective, and algebraic. Furthermore, we situate the points PX within the list of triangle centers.