Apollonius problem in terms of oriented circles

Abstract

The solution of Apollonius' problem on constructing a circle (line), tangent to three given circles (lines), is presented in terms of oriented circles and inversive invariants. Tangency is understood as the coincidence of tangent vectors at the common point, in contrast to counter-tangency. The problem has 0, 1 or 2 solutions. By reversing each of the given circles one by one, we obtain the remaining solutions of the classical non-oriented problem.

Figures (14)

• Figure 1: Configuration with 8 solutions; thick lines are given circles (the dashed thick line shows the reversed circle); thin lines (including the straight line in fragment 1:) are Apollonius' circles (Eq. \ref{['abcd0']})
• Figure 2: Constructions for three lines (Eq. \ref{['3lines']})
• Figure 3: Pencils of circles/lines; the inversion circle is shown dashed
• Figure 4: Configurations with two straight lines
• Figure 5: Configuration with 6 solutions; on fragments 1: and 2: $Q_1=0$, solutions \ref{['abcd0']} are multiples

Theorems & Definitions (1)

• Corollary