It is well known that the eight circles, each of which touches three given circles, 
are determined as follows:—viz. considering any one in particular of the four axes of 
similitude of the given circles, and the perpendicular let fall .from the radical centre 
(or centre of the orthotomic circle) of the given circles, there are two of the required 
tangent circles which have their centres upon the perpendicular, and pass through the 
points of intersection of the orthotomic circle and the axis of similitude, or in other 
words, the axis of similitude is a common chord (or radical axis) of the orthotomic 
circle and the two tangent circles. This suggests the choice of the radical centre for 
the origin of coordinates; and the resulting formulae then take very simple forms, and 
the theorem is verified without difficulty. 
Take then the centre of the orthotomic circle as the origin of coordinates, and 
let the radius of this circle be put equal to unity ; then if (a, /3), (a', /3'), (a", /3") 
are the coordinates of the centres of the given circles, the equations of these will be 
and the radii of the circles will be Va 2 + /3' 2 — 1, Va /2 + /3 /2 — 1, Va ,/2 + /3"~ — 1, It will be 
convenient to write 
y = + V a 2 + /3 2 — 1,