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.

86

A SEQUEL TO EUCLID.

In like manner, the line SS' passes through the
same centre of similitude. Hence the point 0, where
these lines meet, will be the external centre of similitude of X and Y; and.'. the rectangle OR. OR'
= OS. OS' (Prop. 2);.-. tangent from O to W = tangent from O to V, hence the radical axis of W and V
passes through O.
DEF.-The circle on the interval, between the centres of
similitude of two circles as diameter, is called their circle
of similitude.
Prop. 5.-The circle of similitude of two circles is the
locus of the vertex of a triangle whose base is the interval
between the centres of the circles, and the ratio of the sides
that of their radii.
Dem.-When the base and the ratio of the sides are
given, the locus of the vertex (see Prop. 7, Section I)
is the ( whose diameter is the interval between the
points in which the base is divided in the given ratio
internally and externally; that is, in the present case,
the 0 of similitude.
Cor. 1.-If from any point in the ( of similitude of
two given Os lines be drawn to their centres, these
lines are proportional to the radii of the two given (s.
Cor. 2.-If, from any point in the 0 of similitude of
two given Os, pairs of tangents be drawn to both Os,
the angle between one pair is equal to the angle between
the other pair.
This follows at once from Cor. 1.
Cor. 3.-The three Os of similitude of three given
Os taken in pairs are coaxal.
For, let P, P' be the points of intersection of two
of the Os of similitude, then it is evident that the
lines drawn from either of these points to the centres
of the three given Os are proportional to the radii of
the given Os. Hence the third 0 of similitude must
pass through the points P, P'. Hence the Os are
coaxal.
Cor. 4.-The centres of the three Os of similitude
of three given Os taken in pairs are collinear.