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.

105

lished by me in the Quarterly Journal for February,
1861:
" If ABC be a plane triangle, the circle passing through
the feet of its perpendiculars touches its inscribed and
escribed circles."
E
B C
D
Dem.-Let the inscribed and escribed (s be denoted by 0, 0', 0", 0"', the circumscribed ( by X,
and the 0 through feet of is by S. Now, if P be
the intersection of I-s, and if the lower segments of
_Is be produced to meet X, the portions intercepted
between P and X are bisected by the sides of ABC
(Prop. 13, Section I., Book III.). Hence S passes
through the points of bisection, and therefore P is the
external centre of similitude of X and S.
Let DE be the diameter of X, which bisects BC.
Join PD, PE, and bisect them in G and H; then S
must pass through the points G and H; and since GH
is 1I to DE, GH must be the diameter of E; and since >
passes through F, the middle point of BC (see Prop. 5,
Section I., Book IV.), the Z GFH is right. Again,
if from the point D three i-s be let fall on the sides of
ABC, their feet are collinear, and the line of colli