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.

58

A SEQUEL TO EMMAID

Dem. —Join HK, HG, IK, IG; then, because AO is
bisected in K, and AC in H, HK is 11 to CO. In like
manner, HG is I1 to AB. Hence the Z GHK is equal
to the Z between CO and AB;.'. it is a rightL; consequently, the 0 described on GK as diameter passes
through H. In like manner, it passes through I; and
since the Z KDG is right, it passes through D;.'. the
A
C M
ircle through the three points G, H, I, passes through
the two points D, K. In like manner, it may be
proved that it passes through the pairs of points E, L;
F, Ml. Hence it passes through the nine points.
DEF.-The circle through the middle points of the sides
of a triangle is called, on account of the property we have
just proved, " The Nine-points Circle of the Triangle.:'
Prop. 6. —To draw the fourth common tangent to the
two escribed circles of a plane triangle, which touch the base
produced, without describing those circles.
Con.-From B, one of the extremities of the base,
let fall a 1A BG on the external bisector AI of the
vertical Z of the A ABC; produce BG and AI to meet
the sides CA, CB of the / in the points H and I; then
the line joining the points H and I is the fourth common tangent.
Dem.-The As BGA, HGA have the side AG common, and the Zs adjacent to this side in the two As