56 The Three Sections, Tangencies , 



ANALYSIS. 



Let O be the centre of the required circle^ and T>, Ej F 

 the points in which it touches the given circles A^ B^ C. 



Then ED passes through P a centre of similitude of the 

 circles B and A ; and FD passes through Q, a centre of simili- 

 tude of the circles C and A. 



And since the rectangles PD.PE^ QD.QF are of known 

 magnitudes, and that they are respectively equal to PO^ — 

 0D2 and QO^— OD^ .-. PO^— QO^ is of known magnitude and 

 sign, and the locus of O is a known straight line 01 perpen- 

 dicular to PQ, and such that PI^ — QI^ = PO^ — QO^ . 



Now if in OB we have OM = OC, then EM = the radius 

 of circle B ; and BM is of known magnitude ; and the circle 

 having B as centre and BM as radius is known. 



Let N be the other point in which OM cuts this last men- 

 tioned circle ; and suppose the circle CHL passing through 

 the point C and having with the circle MN the line 10 as 

 radical axis. 



Then L being the other point in which OC cuts the circle 

 CLH, we have OC.OL = OM.ON ; hence as OC = OM, it 

 follows that CL is equal the diameter MN, and .-.of known 

 magnitude. But the circle CHL is known; .•. CL is known 

 in position, and hence the point O in which it cuts 10, and 

 therefore the circle DEF is known. 



COMPOSITION. 



Find P a centre of similitude of the given circles A and B ; 

 find Q, a centre of similitude of the given circles A and C ; 

 draw a straight line PD'E' cutting circles A, B^ in dissimilar 

 points D' and E' ; draw a straight line Qid'f cutting the given 

 circles A, C, in dissimilar points d' and /' ; find the point I in 

 the line PQ such that PP _ QI^ = PD'.PE' — Qd' .Qif , and 

 through I draw a straight line IB perpendicular to PQ : draw 

 any radius Be' of the circle B, and from e' in the proper 

 direction on e'B make e'm' = the radius of circle C;'with B 

 as centre and Bm' as radius describe a circle, and produce 

 e'm' to cut it in n' ; through the point C describe the circle 

 CHL which with circle Bm'n' has IB as radical axis, and in 

 it inflect the chord CL = to the diameter ni'n, and let O be 

 the point of intersection of CL with III : then will O be a 

 centre of a required circle. 



