On the Contacts of Circles

Extract

There is a theorem for three circles in a plane, that if three tangents, each of two of these circles, either all transverse or one transverse and two direct, can be drawn to meet in a point, then the three tangents, each of two of the circles, respectively conjugate to those first taken, likewise meet in a point. The theorem was stated by Quidde, with a proof for the necessity of the condition as to the tangents to be taken, in a paper designed to establish Steiner's solution of Malfatti's problem. Casey gives the theorem with omission of the condition for the character of the tangents, as does Salmon, who, however, gives a proof depending on the right choice of certain square roots which enter. Quidde's theorem is stated, accurately, in the Nouvelles Annales, and a simple metrical proof, from the diagram drawn (essentially Quidde's, see 6 below) is given later in the same Journal by Mannheim; this is practically repeated by Hart. Recently, Prof. Neville, emphasizing the necessity of the condition for the character of the tangents, has called attention to Quidde's paper.