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.

24

A SEQUEL TO EUCLID.

of the convention regarding plus and minus: thus, since
+ AC = - CA, we get
AB. CD~+ BC.AD=-CA.BD,
or
AB.CD + BC.AD + CA.BD = 0.
Prop. 8.-Ifeperpendiculars be drawnfrom the angular
points of a square to any line, the sum of the squares of
the perpendiculars from one pair of C
opposite angles exceeds twice the
rectangle of the perpendicularsfrom B
the other pair of opposite angles by
the area of the square. b D
Dem. -Let ABCD be the c
square, L the line; let fall the B A
I s AM, BN, CP, DQ, on L:
through A draw EF 1 to L. Now, L N M P
since the L BAD is right, the sum M P
of the Ls BAE, DAF = one right L, and.. = to the
sum of the Zs BAE, ABE;.-. L ABE =DAF, and
L E = F, and AB =AD;.-. AE = DF.
Again, put AM = a, BE = b, DF = e. The four I s
can be expressed in terms of a, 6, e. For BN = a ~ b,
DQ = a + e; and since 0 is the middle point both of
AC and BD, we have BN + DG = AM + CI?, each being
= twice theifromO. Hence (a+b) + (a + c)=a~CP;
therefore CP = (a + b + c).
Now, BN2Y DQW-2AM.CP=(a +b)2+(a + )2
-2a(a + b +c) = b'+c2 B= E2~+ DF2.
- BE2 + EA2 - BA2 - area of square.
Prop. 9.-If the base AB c
of a triangle be divided in D,
so that mAD = n DB; then
mAC2 + nBC2 - mAD2 + nBD2
+ (m + n) CD2.
Dem.-Let fall the I CE;
then A D E B
mAC2 - m(AD2 ~ DC2 ~ 2AD. DE); (xii.)
nBC2 = n (BD2 + DC2 - 2DB. DE). (xiii.)