63 The Three Sections, Tangencies, 



Since HI and HG are parallel to EEQ, and QPO^ the angle 

 QO right to E = angle HI right to G and is .•. = angle DI 

 right to G or DM right to O or E; and /.a circle can pass 

 through MDEQ. 



But O being a centre of similitude to the circles A and B, 

 the magnitude of OD.OE is known; and /. since OM.OQ := 

 OD.OEj the point M is known. 



Let N be the point in which a tangent to circle A at H 

 cuts POQ. 



The angle NH right to P is = HN right to I and /. =^ 

 DH or DP right to I or M. Plence a circle can pass tlu-ough 

 DH, M, and N. And since PJM.PN is = the known magni- 

 tude PH.PDj it follows that the point N is known. 



Now N being a known point, the tangent NH to circle A 

 is known; and the point F in which PH cuts the circle C 

 similarly to H on circle A is known; as also the other point 

 D in which it cuts circle A. And the point E on circle B, in 

 in which OD cuts it dissimilarly to D on circle A, is known: 

 and hence the circle DEE is kno^^iTi. 



Or, we might determine D by drawing a tangent at D to 

 cut OPQ, &c., because circle MDP touches A at D. 



COMPOSITION. 



Find O a centre of similitude of the circles A and B; find 

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

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

 with O and P. 



Through O draw a straight line to cut the circles A and B 

 in dissimilar points, D' and E'; describe the circle QE'D'; 

 through P draw a line to cut the circle A in points H', D'; 

 through H', D', and the other point M in which the circle 

 Q-E'D' cuts the line OPQ, describe a circle; through the 

 other point N, in which the circle D'H'iVI cuts the line QPO, 

 draw NH a tangent to the circle A; di'aw PH to cut the circle 

 A again in D, and to cut the circle C in E dissimilar to D on 

 A; draw OD to cut circle B in E dissimilar to D on circle A; 

 describe the circle DEE. 



Then is DEE a required circle. 



NOTES. 



This method of solution is intelligibly applicable in a direct 

 manner only when the circle A is finite, and C neither an 



