and Loci of ApoUonius, ^r. 45 



straight line Y)'d' antl tlic common tangent to the circles A 

 and DEF at D, cnt each other in PO the radical axis of the 

 circles DEF and D'E'F'. And for like reasons it is also 

 evident that the intersection of E'e' and the tangent to circle 

 B at E, and also the intersection of Y'f and tangent to circle 

 C at F, are in PO. 



Bnt the intersections of the straight Hnes T>'d', E'e', and 

 F/' A^th PO are known ; .*. the tangents from these points to 

 the respective circles A, B, C, are knoAvn ; hence, the points 

 of contact T), E, F, heing known, the circle DEF is known. 



Or, having fonnd cither point of contact the others 



can be easily determined. Thus for instance when D is 

 found, then E and F are the dissimilar points in which OD 

 and PD cut the circles B and C. 



COMPOSITION. 



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

 centre of similitude of circles A and C ; through D' any- 

 assumed point in circumference of circle A, draw OD' and 

 PD' to cut the circumferences of B and C in the points E' and 

 F' dissimilar to D' on circumference A ; describe the circle 

 D'E'F' and di'aw D'd', E'e', F/', its respective chords of inter- 

 section with the circles A, B, C, to cut the straight line PO 

 in a, b, c. ; draw aD tangent to the circle A ; draw OD and 

 PD to cut the circles B and C in the points E and F dissimilar 

 to point D on circle A ; describe the circle DEF. Then v>ill 

 DEF be a required circle. 



For OD.OE being = OD'.OE', and PD.PF = PD'.PF', it 

 follows that OP is the radical axis of the circles DEF and 

 D'E'F', and therefore that aD is tangent to the circle DEF as 

 well as to circle A at the point D : and hence the circle DEF 

 touch.cs circle A at D. 



And since ODE passes through the point of contact D of 

 the circles DEF and A, and that O is a centre of similitude 

 of circles A and B, and that the points D and E on circles A 

 and B arc dissimilar, .-. the circle DEF touches the circle B 

 in E. And for similar reasons the circle DEF touches 

 circle C in F. 



NOTES. 



It is well to observe that we can find the points D and F 

 from E (E being the point of contact of a tangent from 6 to 



