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.

138

A SEQUEL TO EUCLID.

Let A3C be the given triangle; P, Q, R the given
points. Take on AC any A
arbitrary point D. Join R
PD, intersecting AB in E;D /
then join EQ, intersecting p
BC in F; lastly, join FR,
intersecting AC in D'; then
the two points D, D' will B F/ \
evidently form two homographic divisions on AC,
the two double points of q
which will be vertices of two triangles satisfying the
question.
(3). Being given three points P, Q, R, and two lines
L, L', it is required to describe a triangle ABC having
C equal to a given angle, the vertices A and B on the
given lines L, L', and the sides passing through the given
points.
Solution.-Through the point R draw any line
meeting the two lines L, L' in the points a, b. Join
Pb, and from Q draw
Qa' making the required
angle C with Pb; the /
two points a, a' will
form two homographic
divisions on L, the double
points of which will give /
two solutions of the required question. a p L
(4). To inscribe in a
circle a triangle whose sides shallpass through three given
points.
This is evidently solved like the preceding, by taking
three false positions, and finding the double points of
the two homographic systems of points.
(5). The problem of describing a circle touching three
given circles can be solved at once by the method of taking