Has anyone ever thought of the Descartes problem rediscovered by F. Soddy? Given three mutually tangent circles there is a smaller internal fourth circle tangent to all three. I have not found anywhere the proof to the beautiful formula that is just given to all internet sites as granted, but never proven. I have found a proof on pages 157-158 of Dan Pedoe's Geometry but I admit it is utterly incomprehensible to me (what could the product of two circles mean?)

Do you know of a solution perhaps with the use of trigonometry and plane geometry that I could understand? I presume the Descartes problem is the continuation of the Malfatti triangle, i.e. given three mutually tangent circles calculate the sides of the triangle enclosing all those three circles.

Best regards and keep up your excellent work.

Yours faithfully,


Marios Eleftheriadis (Athens, Greece)
Tel.: +30-6944-677903
E-mail: mele88@otenet.gr