Very obvious fact to some, fascinating to me though.
Consider a triangle. Think of it as three points, where three lines intersect. Imagine that we are only dealing with Rational Points (points whose coordinates are rational numbers)
It appears to me, though I have no proof, that each triangle will have the potential to draw upon it an associated set of three circles, all touching each other at one point. (this is also called “mutually tangent” or even “kissing circles”).
I would imagine this is true for any triangle, including funky skinny thin ones
Even in the case where the triangle is just a line, where all points are collinear. There are three circles, it’s just that the middle circle has a radius of zero. Or, you might say, it has infinite curvature. (Curvature being the inverse of Radius, or 1 divided by Radius)
Well…. what happens when you have three mutually tangent circles? This goes back a long, long way. Apollonius of Perga, Pappus, Viete, Renee Descartes and Elisabeth of Bohemia, Sophie Lie, Soddy, and many others.
The thing about these 3 circles is that you can always find another circle, a fourth circle, that is tangent to the first three.
 The Correspondence of René Descartes: 1643, Verbeek, T.; Bos, E.-J. (Erik-Jan); Ven, Jeroen van de, (2003) Quæstiones infinitæ : publications of the Department of Philosophy, Utrecht University, ISSN: 0927-3395, Publisher: Zeno, The Leiden-Utrecht Institute for Philosophy, Utrecht University Repository in NARCIS
 Problem of Apollonius, including section on Sophus Lie, Wikipedia
 Princess Elisabeth of Bohemia, Wikipedia:
Elisabeth of Bohemia-Palatinate with hunting spear from A Sister of Prince Rupert by E. Godfrey. According from the text the original painting this photo is based off of is in the Library of Bodleian Oxford in the School of Honthorst.
|Date||10 December 2013, 21:23:47|
 Francois Viete, Wikipedia