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.

A SYSTEM OF THREE SIMILAR FIGURES. 189

16. The locus of the inverse of either Brocard point with respect
to a Tucker circle is a right line.
17. If the middle points of the lines AH, BH, CH be A", B",
C", respectively, and the middle points of the sides BC, CA, AB
be A"', B"', C"'; then the Simson's line of any of these six
points, with respect to the triangle A'B'C', passes through the
centres of two of Taylor's'circles.
18. If the orthocentres of the triangles AB'C', HB'C' be P, Q,
respectively, the lines A'P, A'Q are bisected by the centres of two
of Taylor's circles.
19. The Simson's lines of any vertex of the triangle A'B'C',
with respect to the four triangles A"B"C"', B"C"A"', C"A"B"',
A"'B"'C'" pass respectively through the centres of Taylor's
circles.
20. Prove that the intercept which the loci in Ex. 16 make on
any side of the triangle subtends a right angle at either Brocard
point.
SECTION IV.
GENERAL THEORY OF A SYSTEM OF THREE SIMILAR
FIGURES.
Notation.-Let F1, F2, F, be three figures directly
similar; a,, a2, a3 three corresponding lengths; a, the
constant angle of intersection of two corresponding
lines of F, and F3; a2, a, the angles of two corresponding lines of F, and F1, of F1 and F,, respectively; S,
the double point of F, and F3; S, that of F3 and F,;
S, that of F1 and F2. We shall denote also by (0, AB)
the distance from the point 0 to the line AB.
DEF. 1.-The triangle formed by the three double points
S,, 2,, S, is called the triangle of similitude ofF1, F,, F3;
and its circumcircle their circle of similitude.
Prop. 1.-In every system of three figures directly
similar, the triangle formed by three homologous lines is
in perspective with the triangle of similitude, and the
locus of the centre of perspective is their circle of similitude.