Non-Euclidean solutions to the Problem of Apollonius
Traditionally, problems of Apollonius are solved with a pencil, ruler and a compass. These primitive tools allow us to construct solutions using the methods of circular inversion, power of a point in respect to a circle, or dilation. However, another way of solving these problems is by using a set of points with a shared property.
The set of points given by a shared property allows us to divide the problem into smaller parts which we’ll be able to solve more easily. For example, let’s consider the problem PLL (consisting of two lines and a point), where the lines are concurrent, and the point lies outside of them:
To determine the set of points, we need two of the specified objects. This gives us three pairs, which we can use to find the set of points. The solutions to the problem can then be found on the intersections between these sets. In this case, the pairs of two lines (LL) and two pairs of a point and line (PL). The solution for pair (LL) is simple, since we know that the centre of the circle, which is the tangent of the two lines, must lie on the axis of the angle of their intersections.
The set of points for a pair (LL) can be drawn using the ordinary drawing tools mentioned before, meaning it is an Euclidean solution. However, we still need to determine the point set of the pairs (PL). To do this, we construct a parabola, which is defined such that all of its points are equally far away from the parabola‘s focus and directrix. These properties are exactly what the centre of our desired circle must have. Since the last pair is the same as the second, we may simply construct another parabola to find the complete set of points.
In this case, the three sets have exactly two points of intersection. These are the centres of circles tangent to both lines and the point.
The set of points of a given property can be constructed for six pairs – LL, PL, CL (circle-line), PP (two poins), PC (point-circle) and CC (two circles). Of these, PL, CL, PC and CC are non-Euclidean, because they cannot be constructed using the standard tools (with the exception of some special cases). We have now solved one Apollonius problem with the pair PL, so let us move on to the pair CL, which can be solved using a similar method. The circles k_1 and k_2 have their centre randomly chosen on the parabolas k, e.
For PC pairs we must use a hyperbola which has its foci located at a given point and in the circle’s centre. One point on the hyperbola sits between the given point A and the intersection of the circle k_1 and the line AB. The definition of the hyperbola implies that |AI| -| BI| = k. This constant is the radius of the given circle k_1. The hyperbola also has a second curve where |AF| - |BF| = -k:
CC pairs are solved in a very similar way, where we create an additional hyperbola due to solutions that pass between the circles:
