Translation of Gergonne's Géométrie Analitique

This is a translation of Gergonne's paper Recherche du cercle qui en touche trois autres sur un plan; published in the Annales de Mathématiques pures et appliquées, tome VII (1816-1817), no. X, 1er avril 1817; pp. 289-303. (Link)

In this paper, Gergonne demonstrates the applications of analytical geometry (i.e. coordinate geometry with algebra, the approach promoted originally by Descartes with his Géométrie). This approach was criticized for being too cumbersome, being dependent on a particular choice of coordinate system, and so being less general than pure (axiomatic) geometry. To combat this sentiment, Gergonne demonstrates a new solution to Apollonius's problem, derived entirely from algebraic considerations. This was a follow-up to a previous work, Mémoire sur le cercle tangent a trois circles donnés, et sur la sphère tangente a quatre sphères données published in the Mémoires de l'Académie Royale des Sciences de Turin, Vol. 22, 1813-1814 (1816), pp. 20-40. (Link)

The construction he develops can be loosely summarized as follows: For the three given circles, we describe the common tangents between them, taken by pairs (Gergonne implicitly assumes the tangents are the outer tangents). For each pair of tangents, we draw the chords of contact and mark the points where certain of these chords of contact intersect. Then connecting these intersections, we find the tangent points on each circle, which define the desired circle. 

For full details, see the paper itself, and my notes in the StackExchange question. 

This translation was made for user Arne Erikson on History of Math and Science StackExchange. 

A small note: It seems that Gergonne has a sign error when he writes equation (n). The construction is valid, however. It's only his algebra that needs to be clarified. In fact, the analysis is fairly lax (maybe even sloppy) and could really use a more careful treatment.