find Loci of Ajwllonius, ^-c. 47 



ANALYSIS. 



Let O be the centre of tlie required eircle, and let D^, E, F 

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

 OAU, OBE, OCF are straight lines. 



Since OD = OE, if OG bisects the angle OD riffht round 

 to OE, and that AG is perpendicular from A on OG, then is 

 the locus of G a known circle having its centre in the middle 

 point of AB, and such that its diameter jNIP intercepted by 

 AG and OG is parallel to AO. Similarly, since OD = OF, 

 it follows that if Oil bisects the angle OD rir/ht round to OF, 

 and that AH is perpendicular from A on OH, then ^\i\\ the 

 locus of H be a knoAvn circle having its centre equally distaait 

 from A and C, and such that the diameter NO intercepted 

 between AH and OH shall be parallel to AO. 



And if Ave assume any auxiliary circle having A as centre, 

 and that AH. AN' is equal to the square of its radius, then 

 the locus of N' is a known circle N'H'Q' such that H' being 

 the other point in which AH cuts it, the segments cut off by 

 the chords NH and N'H' are similar. And, for like reasons, 

 if on AG we take the point jNI' such that AG.A]M' = the 

 square of the radius of the auxiliary cii'cle, then Avill the locus 

 of ]M' be a circle having with MGP the point A as a centre of 

 similitude, and such that if G' be the other point in which AG 

 cuts it, the segments cut off" from it and circle PG]M by the 

 chords MG and INI'G' are similar. 



Let I be the point in which AO cuts M'N'. 

 Since the angles G and H are right, and that AG.A^VI' = 

 AH. AN' = square of radius of auxiliary circle, it folloAvs that 

 M'N' is pei*pendieular to AO, and that AO.AI is equal the 

 square of radius of auxiliary circle. 



Again, the angle ^M'N' right to A being equal to the angle 

 OA right to G = PM right to G, it is .-. = P'M' right to G', 

 and .-. M'N' touches circle M'G'P' in M'. Similarly, since 

 angle N'A right to M' = OH right to A = QH right to N, 

 it is .'. equal Q'H' right to N', and .-. M'N' touches the circle 

 N'H'Q' at N'. 



Hence as !M'N' is a common tangent to two known 

 circles it is itself knoAvn, and .*. the centre O which is the pole 

 of ]\rN' in respect to the auxiliary circle, is knoAvn, and .'. 

 also the circle DEF. 



And since the radius of the circle PG!M can be taken equal 

 to the half sinn or half difference of the radii of circles A, B, 

 and that the radius of circle NHQ can be taken equal either 



