54) The Three Sections, Tangencies, 



the ratios of the perpendiculars from the poles of the axes of 

 similitude on the radical axes, &c., &c. 



SIXTH SOLUTION. 



(See Plate.) 



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



ANALYSIS. 



Let D, E, and F be the points of contact of the required 

 circle with the three given circles A, B, and C. 



Then DE, DF and JEF pass through O^ P and Q centres of 

 similitvide of the given circles; and OPQ, is an axis of 

 similitude. 



Let G and H be the other points in which ED and FD cut 

 the circle A, and S that in which HG (which is parallel EFQ) 

 cuts POQ. 



Then OS has to OQ. the known ratio which OG has to OE, 

 and which radius A has to radius B, and .-. S is a known 

 point. 



Now the ratio OG.DH to PH.DG, being the same with 

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

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

 ratios, or that of PD.DH to OD.DG is known : or — which 

 amounts to the same — the ratio of OE.FD to PF.ED being 

 the same with that of OQ to PQ. is knov^n ; and the ratio of 

 PF.PD to OE.OD is known; and .-. the ratio compoimded 

 of these ratios, or that of PD.DF to OD.DE is known; and 

 .-., as DF : DE : : DH : DG, it follows that the ratio of 

 PD.DH to OD.DG is known. 



Let I be the other point in which a circle through P, H 

 and G would cut OGD. Then PD.DH = ID.DG; and .-. 

 ID has to OD the known ratio which PD.DH has to OD.DG, 

 and the point I must be in the circumference of a known 

 circle ILL having with circle A the point O as a centre of 

 similitude. Moreover, if K be the other point in which DO 

 cuts this circle, then as the angle LK riglit to I is = HG 

 right to D, it is = ID right to P; and .*. PI is a tangent to 

 the circle IKL at I. 



Now PI is known, and .•. also 10 and the point D on circle 

 A similar to I on circle ILK, as also the point E on B dis- 

 similar to D on A ; and the point F on circle C in which PD 



