48 The Three Sections, Tangencies, 



the half sum or half difference of the circles A^ C, and that to 

 each of the resulting pairs of circles there are two answerable 

 tangents^ .-.it is evident there are four pair of answerable 

 centres O, and .*. eight solutions to the question Avhich are 

 real or unreal in pairs. 



The composition may be easily made. And it may be as 

 well to remark that when we suppose the circle C infinitely 

 grcat^ then will the circle NHQ also be infinitely great ; and 

 its infinite circumference bisects all straight lines drawn from 

 the point A to the infinite circumference of circle C^, &c., &c. 



It may be right to observe that by introducing an auxiliary 

 circle into the Fourth Solution in a similar manner to that 

 in this solution^, we can make it intelligibly applicable to the 

 minor cases which now escape it ; but though this might be 

 an advantage as regards the greater generality obtained^ it 

 would not indicate such neat solutions to the leading cases. 



THIRD SOLUTION. 



(See Plate.) 



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



ANALYSIS. 



Let J), E and F be its points of contact Avith the given 

 circles A, B and C. 



Then DE, DF and EF pass through the respective points 

 O, P and Ql, centres of similitude of the given circles which 

 are in one straight line. 



Let G and L be the other j)oints in which DE cuts the 

 circles A and B ; and let H and M be those in which DF 

 cuts the circles A and C ; and let S be the point in which 

 Gil cuts the axis of similitude OPQ. Then OS has to OQ 

 the known ratio which OG has to OE^ and .•. the point S is 

 known. 



Now the ratio of OG.DH to PH.DG, which is the same as 

 that of OS to PS^ is known; and the ratio of PD.PH to 

 OD.OG is also known ; .• . the ratio compounded of these ratios 

 or that of PD.DH to OD.DG is known : and hence as DH : 

 DG : : DF : DE, it follows that the ratio of PD.DF to OD.DE 

 is known. 



Or — which amounts to the same — the ratio of OE.FD to 

 PF.ED being the same as that of OQ to PQ is known ; and 

 the ratio of PF.PD to OE.OD is (the same with that of 



