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 IV.

61

tangent to the two escribed Os which touch the base
produced, with respect to the ( whose centre is D,
and which cuts these Os orthogonally; if we join D to
the points of contact of the fourth common tangent,
the points where the joining lines meet these Os again
will be the inverses of the points of contact. Hence
they will be common both to the " Nine-points Circle "
and the escribed Os;.. the "Nine-points Circle"
touches these escribed Os in these points; and in a
similar way the points of contact with the inscribed 0
and the escribed ( which touch the base externally
may be found.
Cor. 3.-Since the "Nine-points Circle" of a plane
L is also the "Nine-points Circle" of each of the
three Ls into which it is divided by the lines drawn
from the intersection of its.-s to the angular points,
we see that the " Nine-points Circle" touches also the
inscribed and escribed circles of each of these triangles.
Prop. 8.-The following Propositions, in connexion
with the circle described about a triangle, are very important: —
(1). The lines whichjoin the extremities of the diameter,
which is perpendicular to the base of a triangle, to the vertical angle, are the internal and external bisectors of the
vertical angle.
E
Dem.-Let DE be the diameter L to BC. Join AD,
AE. Produce AE to meet CB in I. Now, from the
construction, we have the arc CD = the arc BD. Hence
the Z CAD = DAB;.. AD is the internal bisector of the
Z CAB. Again, since DE is the diameter of the 0,