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.

BOOK VI.

119

of similitude of the Os 0", 0"'; but this centre is a
point on the radical axis of the circles O, 0' (see Prop. 4,
Section II.). Hence the point E is on the radical axis,
and, by similar triangles,
O"A: "'B:: O"E: O"'E: radius of O": radius of O"',.'. radius of O": O"A: radius of 0"': O"'B;
but the two last terms of this proportion are constant,.'. radius of 0": O"A in a constant ratio.
Prop. 8.-If a chord of one circle be a tangent to
another, the angle which the chord subtends at either limiting point is bisected by the line drawn c
from that limiting point to the point
of contact.
Let CF be the chord, K the point K
of contact, E one of the limiting
points: the angle CEF is bisected by
EK. For since the limiting point E\\
E is coaxal with the circles O, O' we
have, by Prop. I. (3),
CE: CK:: FE: FK;. EC: EF:: K: KF.
Hence the angle CEF is bisected (VI. iii).
In like manner, if G be the other limiting point, the
angle CGF is bisected by GK.
Cor. 1.-If the circles were external to each other,
and the figure constructed, it would be found that the
angles bisected would be the supplements of the angles
CEF, CGF.
Cor. 2.-If a common tangent be drawn to two
circles, lines drawn from the points of contact to either
limiting point are perpendicular to each other; for
they are the internal and external bisectors of an angle.
Cor. 3.-If three circles be coaxal, a common tangent to two of them will intersect the third in points
which are harmonic conjugates to the points of contact;
for the pencil from either limiting point will be a harmonic pencil.