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.

62

A SEQUEL TO EUCLID.

the Z DAE is right;.-. the Z DAE = DAE; and
from these, taking away the equal Zs CAD, DAB, we
have the Z CAE = BAH;;.. JAH = BAH. Hence
AH is the external bisector.
(2). If from D a perpendicular be let fall on AC, the
segments AG, GO into which it divides AC are respectively
the half sum and the half difference of the sides AB, AC.
Dem.-Join CD, GF. Draw EH 1 to AC. Since the
Zs CGD, CFD are right, the figure CGFD is a quadrilateral in a 0. Hence the Z AGF = CDE
(III., xxii.) = CAE (III., xxi.);.-. GF is 1I to AE.
Hence AHFG is a i; and AG = FEH = i sum of AB,
AC (I., 11, Cor. 1). Again, GO = AC - AG = AC
- (AB + AC) = (AC -AB).
(3). If from E a perpendicular EG' be drawn to AC,
CG' and AG' are respectively the half sum and the half
difference of AC, AB.
This may be proved like the last.
(4). Through A draw AL perpendicular to DE. The
rectangle DL. EF is equal to the square of half the sum of
the sides AC, AB.
Dem.-The Ls ALD, EFI have evidently the s
at D and I equal, and the right Zs at L and F are
equal. Hence the Ls are equiangular;.. DL. EF
= AL. FI = FK. FI = the square of half the sum of the
sides (Prop. 7).
(5). In like manner it may be proved that EL. FD
is equal to the square of half the difference of AC, AB.
Prop. 9.-If a, b, o denote, as in Prop. 1, the lengths
of the sides of the triangle ABC, then c
the centre of the inscribed circle will
be the centre of mean position of its
angular points for the system of mul- b / b
tiples a, b, c.
Dem.-Let O be the centre of L
the inscribed 0. Join CO; and on A
CO produced let fall the JLs AL,
BM. Now, the L s ACL, BCM have the Z ACL = BCM;