Every Triangle corresponds to a Descartes configuration of Circles

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

triangle with one tiny leg, two large legs, showing 3 circles drawn with centers at the triangles vertexesa triangle with one long leg, and two smaller legs almost the same length, so that the area of the triangle is quite small. two large circles are drawn at the far vertexes, and a tiny circle rests between them at the middle vertex

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)

a line with three points, two circles drawn with centers at the far points, and a 'zero circle' indicated at the central point. all tangent.

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.

1 2Sophus_Liesod

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 funny thing here is that the both the radius and the center point of the fourth circle will always be rational too! It won’t be a square root or irrational, it won’t be a transcendental number, it will be the ratio of two integers. This is discussed in a fascinating paper on the topic from some guys at the AT&T company a few years back called “Beyond the Descartes circle theorem” by Jeffrey C. Lagarias, Colin L. Mallows, and Allan R. Wilks.

The exception here is the case where the third circle is a zero radius. But then… can we say that the ‘fourth circle’ has a “zero curvature”, in other words, that it is a line? A line tangent to both the ‘big circles’ and also passing through the ‘zero circle’ at a single point? And maybe even consider it is two lines, one with ‘up’ orientation and one with ‘down’?

two tangent circles with a line between them, tangent to both. indicating it can also be considered a circle of zero curvature

To me the interesting bit is that to any triangle in a plane, there is a corresponding set of 3 tangent circles. And vice versa.

But, if you look at it another way, the lines that are triangle’s sides…. are really just circles of zero-curvature. In a sense, a triangle is a set of three circles with zero curvature. Then, the three Descartes circles are a sort of ‘dual’ form of those circles. . . one can transform from the triangle-form to the descartes-circles form and back again, but it could be said that one is really transforming from one set of circles to another. And they are all rational.

My other question is this — if the three Descartes circles can transform back into a triangle, what happens to the ‘fourth’ and ‘fifth’ circle? Do they also ‘transform’ back into something in the plane if you treat them as you treated the other three? What is their ‘dual’ in this way of thinking?

It would be interesting to know how this works in Sophus Lie’s Sphere Geometry, where he has a nice algebra system to describe circles, points, and lines all with one equation. And it’s not the plain old Ax^2 + By^2 + Cx + Dxy + Ey + F = 0.

[1] 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

[2] Problem of Apollonius, including section on Sophus Lie, Wikipedia

[3] 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
Source https://archive.org/details/sisterofprinceru00godf
Author Elizabeth Godfrey

[4] Francois Viete, Wikipedia

[5] “Beyond the Descartes circle theorem” by Jeffrey C. Lagarias, Colin L. Mallows, and Allan R. Wilks. AT&T, via http://arxiv.org

[6] Frederick Soddy, Wikipedia. image from Nobel Prize website


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