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.

121

also the triangles BDE, BD'E'; hence
CD: DB:: C'D': D'B;
DB: DE: D'B: D'E;
therefore, ex aeguali,
CD: DE:: C'D': D'E'.
Cor. 1.-If two lines be divided proportionally, the
circles passing through their point of intersection and
through pairs of homologous points are coaxal.
Cor. 2.-If from the point B perpendiculars be
drawn to the lines joining homologous points, the feet
of these perpendiculars are collinear. For each lies on
the line joining the feet of the perpendiculars from B
on the lines AC, AC'.
Cor. 3.-The circles described about the triangles
formed by the lines joining any three pairs of homologous points all pass through B.
Cor. 4.-The intersection of the perpendiculars of
all the triangles formed by the lines joining homologous points are collinear.
Cor. 5.-Any two lines joining homologous points
are divided proportionally by the remaining lines of
the system.
Prop. 10.-To describe a circle touching three given
circles.
Analysis.-Let X, Y, Z be the three given Os,
ABC, A'B'C' two Os which it is required to describe
touching the three given Os; then, by Cor. 2, Prop. 4,
Section IV., the 0 DEF, which cuts X, Y, Z orthogonally, will be the 0 of inversion of ABC, A'B'C', and
the three Os ABC, DEF, A'B'C' will be coaxal (Cor. 2,
Prop. 1, Section IV.).
Now, consider the 0 X, and the three Os ABC,
DEF, A'B'C'; the radical axes of X and these (Ds are
concurrent (Prop. 4); but two of the radical axes are
tangents at A, A', and the third is the common chord
of X and the orthogonal ( DEE; let P be their point
of concurrence. Again, from Prop. 4, Section II., it
follows that the axis of similitude of X, Y, Z is the