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.

116

A SEQUEL TO EUCLID.

This is Prop. 26, Book III., Section I. The following are important deductions:Cor. 1.-The circle A
described on the line
joining a point A to any
point B in its polar,
with respect to a given
circle, cuts that circle \
orthogonally.
Cor. 2.-The intersection of the -Ls of the
A formed by a pair of B
conjugate points A, B,
with respect to a given circle and its centre 0, is the
pole of the line AB.
Cor. 3.-The polars of any point A with respect to
a coaxal system are concurrent. Por, through A and
through the limiting points describe a (D: this (Cor. 4,
Prop. 2) will cut all the (s orthogonally, and the
polars of A with respect to all the Os of the system
will pass through the point diametrically opposite to
A on this orthogonal (D; hence they are concurrent.
Cor. 4.-If the polars of a variable point with respect
to three given 0 s be concurrent, the locus of the point
is the ( which cuts the three given Os orthogonally.
Prop. 4.-If X1, X2, X3, &c., be a system of coaxal
circles, and if Y be any other circle, then the radical axes
of the pairs of circles Xi, Y; X2, Y; X3, Y, &c., are
concurrent.
Dem.-The two first meet on the radical axis of X,,
X2; the second and third on the radical axis of X2,
X3; but this, by hypothesis, is the radical axis of X1,
X2; hence the Proposition is evident.
Prop. 5.-If two circles cut two other circles orthogonally, the radical axis of either pair is the line joining the
centres of the other pair.
Dem.-Let X, Y be one pair cutting W, V, the other
pair, orthogonally; then, since X cuts W and V orthogonally, the tangents drawn from the centre of X to W