IGRP Episode 12 Seminar 3. Problem 13: 3 intersecting circles

Описание: In this seminar we make one step closer to taking on a fun and a much more challenging problem known as the 3-Circle Problem Of Apollonius by solving a related but a milder version of this problem when we are given 3 distinct, fixed, coplanar circles that intersect at a point and we are asked to construct all circles that are tangent to all these 3 intersecting circles using the Euclidean tools alone (with absolute certainty).

We solve this, preparatory, problem by using exactly the heuristic template that was explained in Episode 10 Seminar 1 and that is adjusted specifically for the machinery of inversive geometry.

To that end, we identify a fruitful center, radius, power and number of inversions of the parent plane that transforms the original, difficult to solve, problem into a problem which is more tractable and which we know how to solve.

Indeed. If in the previous problem we flipped the original, difficult to solve, problem into a problem of drawing "straight lines" tangent to 2 given "circles" then in this problem we flip the difficult to solve problem into a problem of drawing "circles" that are tangent to 3 given and intersecting "straight lines".

It goes without saying that we highly recommend that our viewers trace and shadow all the actual Compass-And-Straightedge constructions that we hint at and carry out on the blackboard by themselves, using their favorite graphing software for this is the only way to put a proper understanding of the subject matter into one's bone marrow.

We also suggest that for the next seminar our students review all the theoretical and practical material and bring their best game because in that seminar we will take on the said 3-Circle Problem Of Apollonius.