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 ~I.

137

If the three pairs of points be on a circle, the points
of intersection of Pascal's line with the circle will be
the double points required. For (see fig., Prop. 5), let
D, B, F; A, E, C, be the two triads of points, and let
the Pascal's line intersect the circle in the points P and
Q; then it is evident that the pencil (A. PDBF)
= (D. PAEC).
Cor.-If we invert the circle into a line, or vice versd,
the solution of either of the Problems we have given
here will give the solution of the other.
Prop. 11. —We shall conclude this Section with the
solution of a few Problems by means of the double points
of homographic division.
(1). Being given two right lines L, L', it is required
to place between them a line AA', which will subtend
given angles 0, 2' at two given points P, P'.
Solution.-Let us take
arbitrarily any point A on A
L. Join PA, P'A, and make
the Zs APA', APA", re- _ AA
spectively equal to the two
given Zs 0, i2'; then,
when the point A moves
along the line L, the points
A', A" will form two ho- \
mographic divisions on the
line L'. The two double points of these divisions will
give two solutions of the required Problem.
(2). Being given a polygon of any number of sides,
and as many points taken arbitrarily, it is required to
inscribe in the polygon another polygon whose sides will
pass through the given points.
We shall solve this problem for the special case of
a triangle; but it will be seen that the solution is
perfectly general,