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.

108

A SEQUEL TO EUCLID.

A'B'C', and is.. = to the radius of Y. Now, let the
radii of X and Y be R, r, and let the distance 00' between their centres be denoted by 3; then we have,
because P is the inverse of E, and Q of F,
01P = 01Q =.
R + 8 ' R - a'
but O'P + 0 OQ = 11Q = r;
r2 rZ
therefore R +
1 1 1
Hence 1+3
R + + i - 3 r'
a result already proved by a different method (see
Prop. 11, Section I.).
Prop. 13. —ITf a variable chord of a circle subtend a
right angle at a fixed point, the locus of its pole is a circle.

Dpem.-Let X be the given circle, AB the variable