How does the Apollonius circle relate to the ellipse?

Authors

  • Jiří Blažek Pedagogical Faculty, University of South Bohemia, České Budějovice
  • Pavel Leischner Pedagogical Faculty, University of South Bohemia, České Budějovice

Abstract

In the article, the equivalence of three characteristic properties of an ellipse used as its definitions is synthetically proved. Circular directrix and auxiliary circle are introduced from the point of view of Circles of Apollonius, the properties of which serve as the basis of subsequent considerations.

The proof of equivalence of the first two properties is derived from the fact that for every point P of an ellipse the colinearity of three points occurs: Its point N on circular directrix, perpendicular projection of the point P on directrix and image of the focus of an ellipse in the circle inversion of circular directrix. This property can be exploited to the construction of points of the ellipse if their distance from the main axis is given.

The equivalence of the third property is proved by the simple use of Pythagorean Theorem and an auxiliary circle.

Published

2019-04-30

How to Cite

Blažek, J., & Leischner, P. (2019). How does the Apollonius circle relate to the ellipse?. MATHEMATICS–PHYSICS–INFORMATICS, 28(2), 81–91. Retrieved from https://mfi.upol.cz/index.php/mfi/article/view/445

Issue

Section

Mathematics