A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

BOOK VI.

85

Again, let us consider the system of points A ', B",
C'. We have, as before,
AC' a
C'B =-b;
BA" b
A"C c;
CB1"
B"a = -'
Hence the product of the ratios in this case also is
negative unity; and.'. A", B", C' are collinear; and
the same holds for A', B", C"; A", B', C". Hence the
collinearity of centres of similitude will be one external and two internal, or three external centres of
similitude.
Cor. 1.-If a variable ( touch two fixed Os, the
line, joining the points of contact passes through a
fixed point, namely-a centre of similitude of the two
Os; for the points of contact are centres of similitude.
Cor. 2.-If a variable 0 touch two fixed Os, the
tangent drawn to it from the centre of similitude
through which the chord of contact passes is constant.
Prop. 4.-If two circles touch two others, the radical
axis of either pair passes through a centre of similitude of
the otherpair.
Dem.-Let the two
Os X, Y touch the
twoOs W,V; letR,B' /
be their points of contact with W, and S, S' Y j o
with V. Now, consider the three s \ s
X, W, Y; R, R' are
internal centres of similitude. Hence the
line RR' passes through the external centre of similitude of X and Y.