and Loci of Apollonius, ^c. 51 



as to employ the steps in a graphic construction ; and that, 

 for this reason, it Avill be compulsory to vary the steps, as is 

 well exemplified in the 3rd solution to the Tangencies. 



FOURTH SOLUTION. 



(Sec Plate.) 



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



ANALYSIS. 



Let O be the centre of the required circle, and let D, E, F 

 be its points of contact with the given circles A, B, C. 



In OB take OH = OA, and then EH is = DA; and it is 

 evident the circle having B as centre and BH as radius is 

 known. Moreover, if M be the other point in which AH 

 cuts this circle, and K that in which a tangent to it at M cuts 

 x\.0, then, since BM and AO are parallels, it follows that JNIK 

 is perpendicular to AO. It is also evident that AO.AK = ^ 

 AH.xVM, and is .•. of known magnitude. 



Similarly, if in OC we take 01 = OA, and that from the 

 other point P in which AI cuts the known circle having C as 

 center and CI as radius, we draw a tangent to cut AO in L, 

 then will this tangent be perpendicular to AO, and will 

 AO.AL = \ the known magnitude AI.AP. 



Now if N be the point in Avliich MK cuts AIP, then as 

 AN.AI has to AP.AI the same ratio which AN has to AP or 

 which AK has to AL or which AK.AO has to AL.AO, it 

 follows that AN.AI = twice AK.AO = AM.AH, and .-. that 

 the locus of N is a knoAvn circle G having with circle C the 

 point A as centre of similitude. 



And since PL is tangent to C at P, it is CAadent KN is 

 tangent to circle G at N ; .*., since MKN is common tangent 

 to the two known circles BM and GN, it is itself kno-\vn ; and 

 AK perpendicular to it is known, as also the point O such 

 that AO.AK = ^ the known magnitude AH.AM. Hence 

 the circle DEF is known. 



COMPOSITION. 



Draw any radius BE' of the given circle B ; draw any 

 radius CF' of the given circle C ; in BE' and CF' make E'H' 

 and F'l' each equal to the radius of the given circle A ; with 

 B and C as centres and BN' and (T as radii describe circlen ; 



E 2 



