and Loci of Apollonius, ^c. 49 



PF.P:\I. rad B to OE.OL. rad C) known; hence the ratio of 

 PD.DF to OD.DE is known. 



Let N be the point in wliich a tangent to the circle C at M 

 cuts tlie straight line ODE. 



It is evident DE.DN = DF.DM, and that DE.DN has to 

 OD.DE a ratio compounded of the known ratios of PD.DF to 

 OD.DE, and of D:\r to PD. 



Or — which amounts to the same — it is evident DE.DN has 

 to OD.DE a ratio compounded of the known ratios of OO to 

 PQ, of PD.PF to OD.OE, and of DM to PD, which may 

 evidently he expressed as the ratio compounded of the ratios 

 of OQ to pa and of PF.DM to OD.OE. 



Hence it follov.s that the point N must be in the circum- 

 ference of a knoAvn circle NXX having with circle A the point 

 O as centre of similitude. 



Moreover, if K be the other point in Avhich DE cuts eii'cle 

 NXX we have the angle XK right to N = angle FE right to 

 D, and .'. = angle ND right to M ; and hence MN is 

 tangent to the circle NXX at N. 



Now MN being a common tangent to two known cii'cles, 

 it is itself known; and .-. the other point F in which the 

 straight line PM cuts circle C is known, as also the point D 

 in wliich it cuts the circle A similarly to the point INI on circle 

 C ; and the point E in which ON cuts the circle B similarly 

 to point N on circle NXX is known: and /. the required 

 circle DEF is kno^ai. 



COMPOSITION. 



Find O, a centre of similitude of A and B ; find P a centre 

 of similitude of A and C ; and Q a centre of similitude of B 

 and C in the line OP ; and find the point S in OPQ such that 

 OS : OQ : : rad xV : rad B. 



Through O draw a straight line OD'E' to cut the circles A 

 and B in dissimilar points D' and E' ; di-aw PD' to cut circle 

 C in the point F' which is dissimilar to D' on circle A, and to 

 cut it again in M' ; find the point N' in OD'E' such that 

 D'N'.D'E' shall have to O'D'.D'E' the ratio compounded of the 

 ratios of D'M' to PD' of OS to PS and of PD.PH to OD.OG, 

 — or which amounts to the same — such that D'N'.D'E' shall 

 have to OD'.D'E' a ratio compounded of the ratios of OQ to 

 PQ and of PF'.D'M' to OD'.OE'; draw N' I parallel D'A to 

 cut OAB in I ; with I as centre and radius IN' describe a 

 circle ; draw MN a common tangent to the circles C and I ; 



E 



