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.

55

Cor. 5.-If we denote the area of the A ABC by N,
we shall have
N = /s (s - a) (s - b) (s - c).
For, by equations (a) and (1i), we have
rs = N, and r' (s - a) = N.
Therefore, multiplying and substituting from (y), we
get
N2 = s (s - a) (s - b) (s - c);
therefore N = /s(s - a) ( - b) (s - c).
Cor. 6.- N = V/r. r'. r". r'";
where r", r"' denote the radii of the escribed circles,
which touch the sides b, c, externally.
Cor. 7.-If the A ABC be right-angled, having the
angle C right,
r=s - c; r=s - b; r"= s - a; rf =s.
Prop. 2.-If from any point perpendiculars be let fall
on the sides of a regular polygon of n
sides, their sum is equal to n times the
radius of the inscribed circle.
Dem.-Let the given polygon be, E C
say a pentagon ABCDE, and P the
given point, and the.Ls from P on
the sides AB, BC, &c., be denoted
by pi, p2, P,, &c., and let the cor- A B
mon length of the sides of the polygon be.; then
2 A APB = spi;
2 \ BPC = sp2;
2 A CPD = sp3;
&c., &c.;
therefore, by addition, twice the pentagon
=S (p1 +P2 +3 +P4 +P6).
Again, if we suppose 0 to be the centre of the inscribed circle, and R its radius, we get, evidently,
2 A AOB = Rs;