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.

83

large circle, whose centre is at infinity in the direction
perpendicular to the line, the centres of similitude of a
line and a circle are the two extremities of the diameter
of the circle which is perpendicular to the line.
Prop. 1.-The direct common tangent of two circles
passes through their external centre of similitude.
Dem.-Let O, O' p
be the centres of the
Os; P, P'the points
of contact of the ( /Or
common tangent; \
and let PP' and 00'
produced meet in T;
then, by similar s,
OT: O'T:: OP: 'P'.
Hence the line 00' is divided externally in T in the
ratio of the radii of the circles; and therefore T is the
external centre of similitude.
Cor. 1.-It may be proved, in like manner, that the
transverse common tangent passes through the internal
centre of similitude.
Cor. 2.-The line joining the extremities of parallel
radii of two Os passes through their external centre of
similitude, if they are turned in the same direction;
and through their internal centre, if they are turned
in opposite directions.
Cor. 3.-The two radii of one ( drawn to its points
of intersection, with any line passing through either
centre of similitude, are respectively 1] to the two
radii of the other (D drawn to its intersections with
the same line.
Cor. 4.-All lines passing through a centre of
similitude of two Os are cut in the same ratio by
the Os.
Prop. 2. —If through a centre of similitude of two circles
we draw a secant cutting one of them in the points R, R',
and the other in the corresponding points S, S'; then
G2