About a sequence of points and a relationship between pencils of conics and circles in the Euclidean plane

TL;DR

This paper explores a geometric relationship involving sequences of points on a triangle's side, connecting pencils of conics and circles in the Euclidean plane, with generalizations to real functions.

Contribution

It introduces sequences of points on a triangle's side and links them to pencils of conics touching perpendicular bisectors, extending classical geometric configurations.

Findings

Centers of circumscribed circles generate a pencil of conics.

Sequences relate to conics touching perpendicular bisectors.

Generalizations to real functions expand classical geometric results.

Abstract

For a given triangle $\triangle ABC$, we define two sequences of points on line $BC$ and provide their generalizations to real functions such that centers of circumscribed circles around $A$ and adjacent points in subsequences generate a pencil of conics touching perpendicular bisectors of $AB$ and $AC$.