How to use Apollonea.com

This website serves as an overview of Apollonian problems and selected methods that can be used to solve them. It contains various variants of the problems according to the relative positions of the given objects, together with interactive constructions created in GeoGebra.

Apollonian problems have a common goal: to construct a circle that satisfies three given conditions. These conditions may be passing through a given point, being tangent to a given line, or being tangent to a given circle. The individual types of problems are therefore denoted by combinations of letters:

• B means point,
• P means line,
• K means circle.

For example, a problem of type BPK means that we are looking for a circle that passes through a given point, is tangent to a given line, and is tangent to a given circle. A problem of type KKK means that we are looking for a circle tangent to three given circles.

The main starting point is the homepage of the website. It contains ten basic types of Apollonian problems according to the combination of three given objects (KKK, KKP, KPP, PPP, BKK, BKP, BPP, BBK, BBP, BBB).

After selecting a basic type of problem, you will either go directly to a specific problem statement or to another navigation page. These additional navigation pages specify the relative positions of the given objects. For example, two lines may be parallel or intersecting, two circles may intersect, be tangent, or lie outside one another, and a point may lie inside a circle, outside a circle, or on a circle. On the website apollonea.cz, you can find a total of 188 different problem variants.

A specific problem page contains the title of the problem, the number of solutions, and a list of available constructions according to the methods used. Some problems are solved by one method, others by two or more methods. Each solution is presented in a separate expandable block. In total, the website contains 388 different solutions of Apollonian problems.

For each problem, there is always at least one Euclidean solution, that is, a solution constructible using a compass and straightedge. Euclidean solutions are marked with a compass icon. Other solutions may also use non-Euclidean objects, such as a parabola or a hyperbola.

After expanding a selected solution, an interactive construction in GeoGebra is displayed. The construction is step-by-step, and the individual steps can be viewed using a slider. The text below the construction describes the solution process step by step.

The construction also includes a Download GeoGebra file link. This file can be opened directly in GeoGebra and further explored there. The user can change the position of the given objects, observe how the construction changes, or analyze the solution in more detail outside the website.

In addition to the problems themselves, the top menu contains two supplementary sections. The section About Apollonian Problems contains more general texts on Apollonian problems, their history, and types of solutions. The section Methods Used contains separate articles on methods that are repeatedly used in the constructions on the website, such as circular inversion, homothety, translation, dilation, power of a point with respect to a circle, or loci.

We hope that exploring Apollonian problems on this website will be both interesting and inspiring.