and Loci of Apollonius, i^c. 55 



cuts it dissimilarly to D on circle A is known. Hence the 

 circle DEF is known. 



COMPOSITION. 



Find O a centre of similitude of the cii'cles A and B ; find 

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

 centre of similitude of the circles B and C which is in line 

 with O and P ; and find the point S in POO, such that VQ, : 

 PS : : rad C : rad A. 



Take any point D' on the circumference of circle A, and 

 draw OD' and PD' to cut the circles B and C in the points 

 E' and F' dissimilar to D' on circle A ; then on OD'E' find 

 the point I' such that I'D' shall have to OD' the ratio com- 

 pounded of the ratios of OS to PS, and of PD.PH to OD.OG : 

 or — which is the same — find I' such that I'D' shall have to 

 OD' the ratio compounded of the ratios of OQ to PQ and of 

 PF.PD to OE.OD. 



DraAv I'M parallel to AD' to cut AO in M ; from M as 

 centre and with MI' as radius describe a circle, to which draw 

 PI a tangent ; draw 10 to cut circle A in D similarly to I on 

 circle M, and to cut circle B in E dissimilarly to D on A ; 

 draw PD to cut circle C in F dissimilarly to circle A in D ; 

 describe the circle DEF. Then is DEF a required circle. 



NOTES. 



Tliis solution holds for all the cases in which the circle A 

 is finite, &c. 



If we were to di'aw DR tangent to the circle A at D to cut 

 PO in R ; then as DR is parallel to PI, and that RP has to 

 RO the known ratio which ID has to OD, it follows that the 

 point R on PO is known ; and /. the tangent RD to circle A 

 is known ; and hence, &c. — another method of solution. 



Or we might solve the problem in a similar manner to that 

 of the third solution from the knowledge that DF.DM has to 

 DE.DL the known ratio of AC^ —(rad A7 rad C)^ to AB^— 

 (rad A ~ rad B)^ where M and L are the other points in which 

 PD and OD cut the circles C and B. 



SEVENTH SOLUTION. 



(Spc Plate.) 



To describe a circle to touch tlu'ee given circles A, B, C. 



