53 The Three Sections, TangencieSy 



draw AH' and AI' to cut these circles again in M' and P' ; 

 find N' in AI' sucIl that AN'.AI' = AH'.AM'; draw N'G 

 parallel P'C to cut AC in G ; with G as centre and GN' as 

 radius describe a circle ; then^ according as ^' has like or 

 unlike sign with ^^ draw MN a common tangent direct or 

 inverse to the circles BM' and GN' ; draw AK perpendicular 

 to MN and in it find the point O such that AO.AK == \ 

 AH. AM (this can evidently be done by producing AK until 

 KA' = AK ; and then describing the circle A'M'H' to cut 

 AK again in O.) 



The point O is a centre of a required circle, &c. 



NOTES. 



Here, too, as in the last solution, it may be remarked that 

 the general solution gives more than one method when applied 

 to many of the particular cases. 



From this solution also we arrive at that given by Cauchy 

 for two circles and a point (by supposing the circle A infinitely 

 small), and we arrive at that of Pappus given in Leslie's 

 Geometrical Analysis for the case of two points and a circle 

 (by supposing the circles A and C infinitely small) . 



Moreover, we see what has not been remarked by the 

 authors of these solutions to the particular cases, viz. : — that 

 the perpendicular from the point A on the tangent MN 

 passes through the centre O of the required circle. 



FIFTH SOLUTION. 



To describe a circle to touch three given circles A,^ B, and 

 C. 



ANALYSIS. 



Let D, E, F, be the points of contact of the required circle 

 with A, B, and C, 



Now (as will appear from some of the porismatic develop- 

 ments), if DN be a perpendicular from D on the radical 

 axis of the circles A and C, and that DM is a perpendicular on 

 the radical axis of the circles A and B, then will DN.AC have 

 to DM.AB one of the four ratios comprehended in that which 

 (AC)2— (rad A ± rad C)^ has to AB^— (rad A ± rad B)2. 



Hence it is evident DN has to DM a known ratio; and .•. 

 as the radical axes UN and RM are known> it follows that 



