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.

57

Dem.-Let 0 be the centre of the 0, then O is the
mean centre of the angular points; hence (II., 10) the
sum of the squares of the lines drawn from P to the
angular points exceeds the sum of the squares of the
lines drawn from 0 by nOP2, that is by nR'2; but all
the lines drawn from 0 to the angular points are equal
to one another, each being the radius. Hence the
sum of their squares is nR2. Hence the Proposition
is proved.
Cor. 1.-If the point P be in the circumference of
the 0, we have the following theorem: —The sum of
the squares of the lines drawn from any point in the circumference of a circle to the angular points of an inscribed
polygon is equal to 2nR2.
The following is an independent proof of this theorem:-It is seen at once, if we denote the I-s from the
angular points on the tangent at P by pi, p2, &c., that
2R. p = AP2;
2R. p2 = BP2;
2R. 3 = CP, &c.
Hence
2R (p1 +P2 +s3 + &c.) = AP2 + BP + CP2, &c.;
or 2R. nR = AP2 + BP + CP2, &c.;
therefore the sum of the squares of all the lines from
P = 2nR2.
Cor. 2.-The sum of the squares of all the lines of
connexion of the angular points of a regular polygon of n sides, inscribed in a 0 whose radius is R, is
n2R2.
This follows from supposing the point P to coincide
with each angular point in succession, and adding all
the results, and taking half, because each line occurs
twice.
Prop. 5.-If 0 be the point of intersection of the three
perpendiculars AD, BE, CF of a triangle ABC, and if
G, F, I be the middle points of the sides of the triangle,
and K, L, M the middle points of the lines OA, OB, OC;
then the nine points D, E, F; G, H, I; K, L, M, are in
the circumference of a circle.