A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Adrian F
Date: 2021 Jan 1, 12:58 -0800
David Pike said on 1st January : "Come on you mathematicians; you must be tempted by that three-circle problem" (quiz no.5)
I’ve looked at the diagram long and hard and I’m struggling to say why the three points are co-linear, but I’ve taken a step into the maths of the problem by showing through calculation that they are, for each of several sets of circle-arrangements. I recognise that’s in a way only re-stating what your diagram shows graphically, but I found it an interesting exercise so thanks for the quiz-question.
As an aside on my method :
Having assigned notional x and y co-ordinates to the centres of the circles, and a radius to each, the x and y co-ordinates of each of points G, E, S (where the tangents meet) were established by trigonometry. I found it useful in this to show that the line going through the centre of each pair of circles (say M and O), when extended also goes through the point S where the tangents meet, and that the lengths MS and OS are in the ratio of the radii.
I used a spreadsheet to calculate the positions of G,E,S for any desired permutation of circle parameters, and the three points are co-linear in every case I’ve looked at.