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.

44

A SEQUEL TO EUCLID.

Prop. 24.-The difference between the squares of the
tangents, from any point P to two circles, is equal to twice
the rectangle contain- E P
ed by the perpendicular from P on the
radical axis and the
distance between the
centres of the circles.
Dem.-LetC, C', \ c D GC C
be the centres, 0 the
middle point of CC',
DE the radical axis.
Let fall the Ls PE, PG. Now,
CP2 - C'2 = 2CC'. OG (II., 6)
CF2 - C'F'2 = CD2 - C'D2,
because DE is the radical axis
= 2CC. OD.
Hence, by subtraction,
PF2- PF12 = 2CC. DG = 2CC'. EP.
This is the fundamental Prop. in the theory of coaxal
circles. For more on this subject, see Book VI.,
Section v.
DEF.-If on any radius of a circle two points be taken,
one internally and the other externally, so that the rectangle contained by their distances from the centre is equal
to the square of the radius; then a line drawn perpendicular to the radius through either point is called the
polar of the other point, which is
called, in relation to this perpendi- P Y
cular, its pole. Thus, let 0 be the
centre, and let OA. OP = radiuss2;
then, if AX, PY be perpendiculars A x
to the line OP, PY is called the 0
polar of A, and A the pole of PY.
Similarly, AX is the polar of P, and
P the pole of AX, -