IGRP Episode 13 Seminar 4. Problem 14. The 3-Circle Problem Of Apollonius

Описание: In this seminar we taken on the 3-circle problem of Apollonius, which, historically, unfolds into a set of 10 related problems the objective of (almost) each of which is to construct circles that are tangent to a mix of straight lines and circles with an additional constraint that the sought-after tangent circles must pass through a given point or points.

These ten related problems are abbreviated as 1) PPP, 2) PPL, 3) PLL, 4) LLL, 5) PPC, 6) PLC, 7) PCC, 8) LLC, 9) LCC and 10) CCC. In order to decipher these abbreviations remember that "P" stands for "a point", "L" stands for "a straight line" and "C" stands for "a circle".

As such, the objective of the PPP problem is to construct a circle that passes through 3 given points. The objective of the PPL problem is to construct a circle that passes through 2 given points and that is tangent to a given (straight) line. The objective of the PLL problem is to construct a circle that passes through a given point and that is tangent to 2 given (straight) lines and so on,

In Episode 11 Seminar 2 we solved the seventh problem of Apollonius, PCC.

In this seminar we solve the tenth problem of Apollonius, CCC. Keep in mind that fact that, evidently, this problem is very rich with history and with mathematics and that it is impossible to do justice to such a problem in one episode.

Consequently, in one episode we have the time to only scratch the surface on this problem and look at the bare-bones technical essence of its inversive geometry-base solution, leaving the discussion of its various other solutions and comparing the textures of these solutions outside of the scope of this course.

We do so by looking at 1) the overall structure of the solution circles of the 3-circle problem of Apollonius (5:19); 2) the 11 essential configurations of its input circles (16:37) and 3) the various algorithms that produce the solution circles of this problem that were covered in the "solution structure" discussion (38:15).

Before deep-diving into the explanation of these algorithms, we explain the two building blocks on which these algorithms are based (27:44 and 32:43).