AN ELEMENTARY SOLUTION TO APOLLONIUS PROBLEMS AND THE INTERSECTION OF CONICS WITH A LINE

Abstract

A solution to the problems of Apollonius is proposed based on the concepts of elementary geometry. It is shown that the proposed solutions also allow addressing another important construction problem: constructing the intersection points of a line and conics (ellipse, hyperbola, and parabola). To solve this problem, conics are considered as a set of points equidistant from two circles.

The presented method emphasizes synthetic geometry tools—namely, compass and straightedge constructions—and avoids reliance on coordinate geometry or algebraic computations. The approach offers a clear geometric interpretation of both Apollonius’ problems and line-conic intersection tasks within a unified framework. This not only provides elegant solutions but also deepens the understanding of geometric relationships through visual reasoning.

In particular, the method enables the construction of conic sections by redefining them as loci of points equidistant from two given circles. This geometric redefinition serves as an alternative to traditional conic definitions involving foci and directrices and enhances pedagogical accessibility for students and educators.

The dual application of this construction—solving both the classical Apollonius problems and the intersection of a line with conics—demonstrates the flexibility and power of elementary geometry. It is especially useful in educational contexts where the development of spatial reasoning and geometric intuition is prioritized.

Overall, the proposed solution contributes to classical geometry by providing a visually intuitive, logically consistent, and broadly applicable method for solving a family of historically significant geometric problems.