46 The Three Sections, Tangencies, 



circle B) by drawing OE to cut the circle A in the point D 

 dissimilar to the point E on circle B^ and then PD to cut the 

 circle C in F the point dissimilar to D on circle A. And in 

 like manner Ave may find D and E from F the point of contact 

 of the tangent from c to cu'cle C. 



INIoreover^ it is e^ddent that we can find D even when the 

 circle A is infinitely great^ for the tangents from a to circles 

 A and D'E'F' are equal. 



As there are two points O, and two points P, there are four 

 lines PO and /. e^ddently four answerable points a; and 

 hence^ as there are two tangents from each point a to the 

 circle A^ it follows that there are in all eight answerable 

 circles DEF^ real or imaginary in pairs. 



Now DD, EE, and FF^ are the polar chords of the circles 

 Aj B, and C in respect to the points a, b, c in PO. Let Q be 

 the centre of similitude of the circles B and C through which 

 EF passes. Then OPQ is straight. And as DD must pass 

 through a' the pole of OPQ in respect to circle A, and that 

 EE must pass through b' the pole of OPQ in respect to circle 

 Bj and that FF must pass through c' the pole of OPQ in 

 respect to circle C ; hence it follows^ because ODE^ PDF and 

 QFE are straight lines^ that T>J), EE and FF must meet in 

 the radical centre II of the giAcn circles A^ B^ C. And this 

 indicates the method of solution given by Gergonne. 



It is also easy to see that DD passes through A' the 

 extremity of the diameter AA' of the known circle D'A^' ; 

 and EE evidently passes through B' the extremity of the 

 diameter BB' of circle BE'e' ; and FF passes through C the 

 extremity of the diameter CC of the circle CF/'. — Hence 

 other methods of solution. 



We may also remark, that as there are four points a, and 

 that the four polars of these points in respect to the circle A 

 pass through the radical centre B of the three given circles, 

 it follows that the four points a are situated in the polar of R 

 in respect to the circle A. 



And, similarly, the four points b are on the polar of B in 

 respect to the circle B, and the four points c are on the polar 

 of E in respect to circle C. 



SECOND SOLUTION. 



iScc Plate.) 



To describe a circle to touch three given circles A. B, C. 



