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.

80

A SEQUEL TO EUCLID.

The method of construction derived from this analysis is evident.
Cor.-If the base, the perpendicular, and the difference of the sides be given, a slight modification of the
foregoing analysis will give the solution.
(3). Given the base of a triangle, the vertical angle, and
the bisector of the vertical angle, to construct the triangle.
Analysis.-Let ABC be the required A, and let the
base AB be given in position; then, since AB is given
in position and magnitude, and the Z ACB is given in
magnitude, the circumscribed (
is given in position. Let CD, /
the bisector of the vertical,/
meet the circumscribed 0 in E,
then E is a given point. Hence
EB is given in magnitude. A
Now ED. EC - EB2 (III.,
20, Cor. 2);.-. the rectangle
ED. EC is given, and CD is
given (Hyp.). Hence ED, EC E
are each given, and the ( described from E as centre,
with EC as radius, is given in position. Hence the point
C is given, and the method of construction is evident.
Cor.-From the foregoing we may infer the method
of solving the Problem: Given the base, vertical angle,
and external bisector of the vertical angle.
(4). Given the base of a triangle, the difference of the
base angles, and the difference of D
the sides, to construct it.
Analysis.-Let ABC be the G
required A; then the rectangle EF. GD = the square of
half the difference of the sides
(IV., 8);.-. EF. GD is given;;
and EF. FD = FB2 is given.
Hence the ratio of EF. GD:EF. FD is given. Hence
the ratio of FD: GD is given.
Again, the L CED = half the difference of the base Z s,