The Apollonius problem for two objects and study of it
https://doi.org/10.22405/2226-8383-2024-25-3-158-176
Abstract
It is well known that the classical Apollonius’s problem to construct a circle tangent to three given circles using a compass and straightedge has finite number of solutions or has no solutions if the given circles are concentric. The so called degenerate cases are also included in the consideration: any of the circles may be a point (a zero-radius circle) or a straght line (a circle of infinite radius).
In this paper we consider the Apollonius problem not for three circles but for only two, with the degenerate cases also considered. We classify all cases of the problem for all possible objects
(points, lines or circles) and for all cases of their mutual arragements on the real coordinate
plane. For every case not only all solutions are provided but also some of their interdependencies
are shown.
The approaches for solutions of the classified cases are based on the notion of locus of the points being equidistant from the given objects and on the equity of distances from the center of the sought tangent circle to each of the given objects.
Unlike the classical Apollonius’s problem the solution always exists, moreover, the number of solutions is infinite.
About the Authors
Anna Sergeevna Kashina
SBEI School № 597 “New Generation"
Russian Federation
Russian Federation
Liliya Mikhailovna Tsybulya
Moscow Pedagogical State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
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Review
For citations:
Kashina A.S., Tsybulya L.M. The Apollonius problem for two objects and study of it. Chebyshevskii Sbornik. 2024;25(3):158-176. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-3-158-176
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