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.

OO00K III.

45

Prop. 25.-If A and B be two points, such that the
polar of Apasses through B,
then the polar of B passes P B D
through A.
Dem.-Let the polar of
A be the line PB; then PB
is L to CP (C being the
centre). Join CB, and let R
fall the -L AQ on CB.
Then, since the Z s P and
Q are right Zs, the quadrilateral APBQ is inscribed in a 0;.. CQ. CB
= CA. CP = radius2;.-. AQ is the polar of B.
Cor.-In PB take any other point D. Join CD, and
let fall the perpendicular AR on CD. Then AQ, AR are
the polars of the points B and D, and we see that the line
BD, which joins the points B and D, is the polar of the
point A; the intersection of AQ, AR, the polars of
B and D. Hence we have the following important
theorem:-The line of connexion of any two points is
the polar of the point of intersection of their polars;
or, again: The point of intersection of any two lines is
the pole of the line of connexion of their poles.
DEF.-Two points, such as A and B, which possess the
property that the polar of either passes through the other,
are called conjugate points with respect to the circle, and
their polars are called conjugate lines.
Prop. 26.-If two circles cut orthogonally, the extremities of any diameter of either are conjugate points with
respect to the other. c
Let the Os be ABF and _
CED, cutting orthogonally in
the points A, B; let CD
be any diameter of the ) 0
CED; C and D are conjugate
points with respect to the -
Q ABF. D