Problems of Apollonius' and Pappus'

The problems of Apollonius date from the 3rd and 2nd centuries BC, when they were formulated by Apollonius of Perga. Originally, it was a single problem: find a circle that is tangent to three given circles. Over time, variations were formulated where the three given objects could include any combination of lines, circles and points. This resulted in ten possible combinations. The goal is always to find the circle that touches all three given objects.

Apollonius' problems were not completely lost thanks to Pappus of Alexandria, but he also created his own specific type of problems – Problems of Pappus. One of the three given objects is a point that also lies on one of the other two objects (a circle or a line).

Although the problems were solved by the author himself, new solutions of the original problem with three given circles, which is the most difficult one, has long been a debated problem in mathematics, and has been addressed by eminent mathematicians such as Isaac Newton, René Descartes, Leonhard Euler, Carl Friedrich Gauss, and many others.

Euclidean geometry provides solutions to all variants of the Apollonian and Pappus problems, using methods such as homothety, dilation, and circular inversion. However, we can simplify the solution with non-Euclidean methods that allow us to construct hyperbolas, parabolas and ellipses. An overview of all problem types, solution procedures and explanations of the methods can be found on this website. Although it is also possible to solve these problems algebraically, this site focuses mainly on geometric methods.