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.

26

A SEQUEL TO EUCLID.

Now, if the point P coincide with 0, OP vanishes, and
we have
A02 + 130 + 002 + D02 = AM2 + MB2
+ 2MN2 + NO2 ~ 3N02 + OD2;
therefore, A]2 + BP2 + CP2 ~ DP2
exceeds AG2 +1 B02 + C02 + D02 by 40P2.
Cor.-If 0 be the point of intersection of bisectors
of the sides of a L, and P any other point; then
AP2 + BP23+ CP2 - A02 + B02 + C02 + 30P2:
for the point of intersection of the bisectors of the sides
is the centre of mean position.
Prop. 11.-Thae last Proposition may be generalized
thus: if A, B, 0, D, &c., be any system of points, 0 their
centre of mean position for any system of multip les a, b,
O, d, &c., then
a. AP2 + b. BP2 + c. CP2 + d.B DP2, &c.,
exceeds a.A02 + b.B"02+Co.00~d. D02P, &C.)
by (a + b + c + d, &c.) OP'.
The foregoing proof may evidently be applied to this
Proposition. The following is another proof from
Townsend's Modern Geometry:From the points A, 3, 0, D, &c., let fall I s AA',
11', CC', PB', &c., on the line OP; then it is easy to
see that 0 is the centre of mean position for the points
A', B', C', B', and the system of multiples a, b, o, d, &c.
Now we have by Props. xii., xiii., Book ii.,
AP2 = AO" + OP2 + 2A'0. OP;
BP2 = B02 + OP2 + 2B'0. OP;
P2 -= C02 + OP2 + 2C'O. OP;
DP2 = D02 + OP2 + 23'0. 0], &c.;
therefore, multiplying by a, b, c, d, and adding, and
remembering that
a. A'0 + b. B'0 + c. 'O + d. 'O + &c. = 0 (see I., 18),