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 II.

23

Prop. 5.-Three times the sum, of the squares of the
sides of a triangle, is equal to four times the sum of the,
squares of the lines bisecting the sides of the triangle.
Dem.-Let A3 E, F be the middle points of the sides.
Then AB' + AC2 = 213D2 ~ 2DA2; (2)
theref ore 2AB2 ~ 2AC2 = 4BiD2 + 41DA2;
that is 2AB 2+ 2AC2 = BC2 + 4DA2.
Similarly 21302 + 2BA2 = CA2 + 4EB2;
and 2CA2 ~20132 = AB2 + 4FC2.
Hence 3(AB2 + 1302 + CA2) = 4(AD 2 ~ BE2 + OF2).
Cor. If G be the point of intersection of the bisectors of the sides, 3AG = 2AD; hence 9AG2=- 4A1D2;
3 (AB2 B C2~+CA2) = 9(AG2 + BG2 +CG2);
(AB' + BC2 + CA2) = 3 (AG2 + BG2 + CG2).
Prop. 6.-The rectangle contained by the sum and
difference of two sides of a triangle is equal to twice the
rectangle contained by the base, and the intercept between
the middle point of the base and the fot of the _perpni
cular from the vertical angle on the base (see Fig., Prop. 2).
Let CE be the.1. and 13 the middle point of the
base AB.
Then AC2 = AE2, + EC2,
and BC2-=BE2 + EC2;
therefore, AC2 - BC2 = AE2 - EB2;
or (AC B C) (AC - BC) =(AE~+EB) (AE - EB).
Xow, AE +EB =AB, and AE -EB =2ED;
therefore (AC + BC) (AC - BC) = 2AB. ED.
Prop. 7.-IfA, B, C, 13 befourpoints taken, in order on
a right line, then AB. CD A BD
~130. AD =AC. BD.
Dem. —Let AB = a, BC =b, CD = 0; then AB. CD
+ BC.AD = acFb (a +b +c) =(a +b) (b +c) =AC.BD.
This theorem, which is dlue to Euler, is one of the
most important in Elementary Geometry. It may be
written in a more symmetrical form by making use