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.

8

A SEQUEL TO EUCLID.

must be always on the line CP, the locus of C is a right
line 1- to the base.
Cor.-The three is of a / are concurrent. Let
the JLs from A and B on the opposite sides be AD and
BE, and let 0 be the point of intersection of these I-s.
Now, AC2 - AB2 = OC2 - OB; (10)
and AB2 - BC2 = OA2- OC2;
therefore AC2 - BC2 = OA2 - OB2.
Hence the line CO produced will be J to AB.
Prop. 11.-If perpendiculars AE, BF be drawn from
the extremities A, B of the base of a triangle on the internal bisector of the vertical angle, the line joining the
middle point G of the base
to the foot of either perpendicular is equal to half the
difference of the sides AC,
BC. D
Dem.-Produce BF to F
D; then in the Ls BCF, G \
DCF there are evidently A \B
two Zs and a side of one
= respectively to two Zs
and a side of the other;.-. CD = CB and FD = FB;
hence AD is the difference
of the sides AC, BC; and, since F and G are the
middle points of the sides BD, BA;.~. FG = I AD
= a (AC - BC). In like manner EG = ~ (AC - BC).
Cor. 1.-By a similar method it may be proved that
lines drawn from the middle point of the base to the feet
of iJs from the extremities of the base on the bisector of
the external vertical angle are each = half sum of AC
and BC.
Cor. 2.-The Z ABD is=ldifference of the base angles.
Cor. 3.-CBD is = half sum of the base angles.
Cor. 4.-The angle between CF and the I from C
on AB = a difference of the base angles.
Cor. 5.-AID = difference of the base angles.
Cor. 6.-Given the base and the difference of the
sides of a A, the locus of the feet of the IJs from the