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 III.

47

= DEF;.-. the Z EFC = EFD. Hence each is right.
Now, since the A OCE is right-angled at C, and CF
perpendicular to OE, OF. OE = OC2; but since the
quadrilateral AFEB has the opposite angles B and F
right angles, it is inscribed in a 0. The rectangle
OF.OE = OA. OB; but OF. OE = OC2;.'. OA. OB
= O02 = radius2;.. BE is the polar of A, and this is
the locus of the point E.
Cor. 1.-If from every point in a given line tangents be drawn to a given circle, the chord of contact
passes through the pole of the given line.
Cor. 2.-If from any given point two tangents be
drawn to a given circle, the chord of contact is the
polar of the given point.
Prop. 29.-The older geometers devoted much time
to the solution of problems which required the construction of triangles under certain conditions. Three
independent data are required for each problem. We
give here a few specimens of the modes of investigation
employed in such questions, and we shall give some additional ones under the Sixth Book.
(1). Given the base of a triangle
the vertical angle, and the sum of /
the sides: construct it.
Analysis.-Let ABC be the <
A; produce AC to D, and make
CD = CB; then AD = sum of
sides, and is given; and the A B
Z ADB = half the Z ACB, and is given. Hence we have
the following method of construction:-On the base AB
describe a segment of a ( containing an Z = half the
given vertical Z, and from the centre A, with a distance
equal to the sum of the sides as radius, describe a (
cutting this segment in D. Join AD, DB, and make
the Z DBC = ADB; then ABC is the A required.
(2). Given the vertical angle of a triangle, and the segments into which the line bisecting it divides the base:
construct it.