arXiv subjects

Miłosz Płatek

Publications and source records attributed to Miłosz Płatek.

5 recordsLinked to original sources

arXiv2026

When Points Become Circles: The Seven Circles Theorem for Arbitrary Closed Six-Circle Chains

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 circle are concurrent. We extend this result to an arbitrary closed six-circle chain. Given such a chain, we consider two circles, each tangent to one of the two alternating triples of circles in the chain. For each circle in the chain, we then construct a circle tangent to it and to both of these circles. We prove that the three lines joining the centers of the newly constructed circles corresponding to opposite members of the chain are concurrent. In the classical configuration, the two circles associated with the alternating triples coincide with the common circle, while the six newly constructed circles degenerate to the six points of tangency, viewed as circles of radius zero. Thus, the classical Seven Circles Theorem is recovered as a degenerate case of our generalization.

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arXiv2026

A New Lemoine-Type Circle Construction

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 the resulting circle to previously known Lemoine circles, in particular the circle introduced by Q. T. Bui. We also note that the resulting circle does not belong to the family of Tucker circles.

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arXiv2026

On the Inscribed Sphere and Concurrent Lines through the Centers of Apollonius Spheres in $\mathbb{R}^n$

The Apollonius problem asks for a sphere tangent to $n+1$ given spheres or hyperplanes in $\mathbb{R}^n$. This problem has been widely studied for an isolated configuration of $n+1$ spheres. In this paper, we study relations among the solutions of the Apollonius problem arising from a common family of spheres within the framework of Lie sphere geometry. More precisely, we consider a configuration of $n+2$ spheres in $\mathbb{R}^n$ and the solutions of the Apollonius problem corresponding to all its subsets of size $n+1$. The first main result concerns lines passing through the centers of pairs of solutions to the Apollonius problem. We prove that all these lines intersect at a single point $P_X$. We then introduce a two--step construction of further Apollonius spheres and show that the lines determined by their centers also pass through $P_X$. This yields numerous applications in two and three dimensions and, at the same time, automatically extends them to $\mathbb{R}^n$. The second main result is an $n$--dimensional generalization of K. Morita's three-dimensional theorem on the inscribed sphere in a configuration of mutually tangent spheres. We show that Morita's theorem is a special case of our result for an arbitrary configuration of $n+2$ spheres in $\mathbb{R}^n$, not necessarily mutually tangent. Moreover, we connect this result with the preceding ones by proving that the center of the corresponding inscribed sphere is again the point $P_X$.

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arXiv2026

Proof of the Generalization of the Sawayama-Thébault Theorem

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 in Laguerre geometry, which provides a natural framework for resolving the problem.

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arXiv2025

A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral

We consider the following configuration. Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, and for each vertex $X$, let $H_X$ be the orthocenter of the triangle formed by the other three. Then $A,\;B,\;C,\;D,\;H_A,\;H_B,\;H_C,\;H_D$ all lie on a single conic. In this paper we study a certain generalization of this fact as follows. For an arbitrary point $P_D$ on the Euler line of $\triangle ABC$, we define corresponding points $P_A, P_B, P_C$ on the respective Euler lines such that the ratio $P_XH_X : P_XO$ is constant for all $X$. We show that the four vertices $A,B,C,D$ and the four isogonal conjugates $Q_A,\;Q_B\;,Q_C\;,Q_D$ of the points $P_X$ all lie on a single conic. This result is given distinct treatments, synthetic, projective, and algebraic. Furthermore, we situate the points $P_X$ within the list of triangle centers.

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