5 papers
When Points Become Circles: The Seven Circles Theorem for Arbitrary Closed Six-Circle Chains
Miłosz Płatek
The Seven Circles Theorem states that if six circles form a closed chain and are tangent to a common circle, then the three lines joining opposite points of tangency on the common…
On the Inscribed Sphere and Concurrent Lines through the Centers of Apollonius Spheres in
MiÅosz PÅatek
The Apollonius problem asks for a sphere tangent to given spheres or hyperplanes in . This problem has been widely studied for an isolated configuration of $n+1…
A New Lemoine-Type Circle Construction
MiÅosz PÅatek
This paper presents a new Lemoine-type construction arising from a configuration of six concyclic points. We prove the concyclicity result, establish a converse theorem, and relate…
Proof of the Generalization of the Sawayama-Thébault Theorem
MiÅosz PÅatek
We prove two conjectures posed in 2016 concerning a generalization of the Sawayama-Thébault Theorem and the Sawayama Lemma. We show that this generalized statement can be viewed i…
A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral
Kazimierz Chomicz, MiÅosz PÅatek, Konstanty Smolira +1
We consider the following configuration. Let be a cyclic quadrilateral with circumcenter , and for each vertex , let be the orthocenter of the triangle formed by…