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.

143

Prop. 2.-If two variable points A, A', one on each of
two lines given in position, subtend an angle of constant
magnitude at a given point 0, the locus of the pole of the
line AA' with respect to a given t
circle X, whose centre is 0, is a
circle.
Dem.-Let AI, A'I be the B
lines given in position, and let
B, B', Q be the poles of the
three lines AI, A'I, and AA' o B'
with respect to X; then the
points B, B' are fixed, and the
lines BQ, B'Q are the polars of
the points A, A';.'. the lines
OA, OA' are respectively I to the lines BQ, B'Q;
hence the Z BQB' is the supplement of the Z AOA';
therefore BQB' is a given angle, and the points B, B'
are fixed; therefore the locus of the point Q is a
circle.
Prop. 3.-For two homographic systems of points on
two lines given in position there exist two points, at each
of which the several pairs of corresponding points subtend
equal angles.
Dem.-Let A, A' be two correspending points on the lines AI, A'I;
and let O, O' be the points on the A
lines AI, A'I which correspond to
the points at infinity on A'I, AI /
respectively; then (see Prop. 10,
Section IV., Book VI.) the rectangle o
OA. O'A' = constant, say P2. Join
OO', and describe the triangle OEO' (see Prop. 15,
Section I., Book VI.) having the rectangle OE. O'E of
its sides = 2, and having the difference of its base Z s
equal difference of base s of the L OIO'. Then E,
the vertex of this A, will be one of the points required. For it is evident from the construction that
OE. O'E = OA. O'A, and that the Z AOE = A'O'E;