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.

60

A SEQUEL TO EUCLID.

Cor. 2.-Let DG cut AB in M, and HII in K, and
from A let fall the J1 AL, then the quadrilateral
LMKI is inscribed in a circle.
For, since the Zs ALB, AGB are right, ALBG is
a quadrilateral in a 0, and M is the centre of the (;.M. XL = MB, and Z MLB = MBL. Again, Z MKI
=AHI = ABI;.-. MKI + MLI = ABI + MBL = two
right Z s. Hence MKIL is a quadrilateral inscribed
in a circle.
Prop. 7.-The "V Nine-points Circle " is the inverse of
the fourth common tangent to the two escribed circles which
touch the base produced, with respect to the circle whose
centre is at the middle point of the base, and which cuts
these circles orthogonally.
Dem.-The Z DML (see fig., last Prop.) = twice
DGL (III. xx.); and the Z HIL = twice AIL; but
DML = HIL, since MKIL is a quadrilateral in a 0;.'. the Z DGL = GIL. Hence, if a 0 be described
about the L GIL it will touch the line GD (III. xxxii.);.'. DL. DI = DG2;.*. the point L is the inverse of the
point I, with respect to the 0 whose centre is D and
radius DG. Again, since MKIL is a quadrilateral in a
0, DM. DK = DL. DI, and,.. = DG2. Hence the point
M is the inverse of K, and.-. the ( described through
the points DLM is the inverse of the line HI (III. 20);
that is, the " Nine-points Circle" is the inverse of the
fourth common tangent, with respect to the ( whose
centre is the middle point of the base, and whose radius
is equal to half the sum of the two remaining sides.
Cor. 1.-In like manner, it may be proved that the
" Nine-points Circle " is the inverse of the fourth common tangent to the inscribed 0 and the escribed 0,
which touches the base externally, with respect to the
( whose centre is the middle point of the base, and
whose radius is = to half the difference of the remaining sides.
Cor. 2. —The " Nine-points Circle" touches the inscribed and the escribed circles of the triangle.
For, since it is the inverse of the fourth common