An elementary treatise on cubic and quartic curves, by A. B. Basset.

CIRCLES OF INVERSION.

145

where R is the radius of the circle circumscribing ABC. Similarly
r2 = 4R2 cos A sin B sin C.
r22= 4R2 cos B sin C sin A,
r32 = 42R cos C sin A sin B.
But OA2 - 2 = 4R2 cos A (cos A + cos B cos C)
= 4R2 cos A sin B sin C = r12.
Similarly OB2- 82= r2,
OC2 - 2= r32
which shows that the circle whose radius is 8 cuts each of the
circles whose radii are rl, r2, r3 orthogonally. In the same way it
can be proved that each of the other four circles cuts every other
one orthogonally.
(v) The radical axis of any two of the four circles passes
through the centres of the remaining two.
The radical axis of two circles is perpendicular to the line
joining their centres; also since the tangents to the two circles
from any point on the radical axis are equal, it follows that if from
any point on the radical axis as centre a circle be described whose
radius is equal to the tangent from this point to either of the two
circles, the last-mentioned circle will cut the first two orthogonally.
Hence the radical axis of the circles whose centres are A and B
passes through the points 0 and C.
214. To find the equations of the four circles.
Let S= 0, U=0, V= O, W= O be the equations of the four
orthogonal circles whose centres are 0, A, B and C. Since ABC
is self-conjugate to S,
S = aa2 cos A + b/32 cos B + cy2 cos C,
also since the sides of the triangle of reference are the radical axes
of S and U, V, W respectively,
7U=S-laI, V=S-m/g3, W = S-nyI......(2.3),
whence la-m/3=0, m/3-n7=0, n7-la =0
are the radical axes of U and V, V and W, W and U respectively.
But these are the equations of CF, AD and BE; whence
I sec A = m secB = n sec C= k,

B. C.

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