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.

92

A SEQUEL TO EUCLID.

0 be the centre of X. Join OA, OB, OC, OD, and let
fall the Ls PA', PB', PC', PD' on these lines; then,
by Prop. 25, Section I., Book III., PA', PB', PC', PD'
are the polars of the points A, B, C, D; and since the
angles at A', C', B', D' are right, the 0 described on OP
as diameter will pass through these points; and since the
system A, B, C, D is harmonic, the pencil (0. ABCD)
is harmonic; but the angles between the rays OA, OB,
OC, OD are respectively equal to the angles between
the rays PA', PB', PC', PD' (III., xxi.). Hence the
pencil (P. A'B'C'D') is harmonic.
DEr.. —Four points in a circle which connect with any
fifth point in the circumference by four lines, forming a
harmonic pencil, are called a harmonic system of points
on the circle.
Prop. 9.-If from any point two tangents be drawn
to a circle, the points of contact and the points of intersection of any secant from the same point form a harmonic
system of points.
Dem.-Let Q be the point, QA,
QB tangents, QCD the secant;
take any point P in the circumfe-
rence of the ), and join PA, PC, A \E B
PB, PD; then, since AB is the
polar of Q, the points E, Q are
harmonic conjugates to C and D;.-. the pencil (A. QCED) is har- /
monic; but the pencil (P. ACBD) \
is equal to the pencil (A. QCED),
for the angles between the rays of one equal the angles
between the rays of the other; therefore the pencil
(P. ACBD) is harmonic. Hence A, C, B, D form a
harmonic system of points.
Cor. 1.-If four points on a ( form a harmonic
system, the line joining either pair of conjugates
passes through the polo of the line joining the other
pair.
Cor. 2.-If the angular points of a quadrilateral
inscribed in a 0 form a harmonic system, the rectangle