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 I.

3

(xxxiv.), because the figures ADFE, CEDF, BFED,
are s.
Cor. 2.-If through the points D, E, any two | s be
drawn meeting the base BC in two points M, N, the
- DENM is = A ABC. For DENM = DEFB
(xxxv.).
DEF.- When three or more lines pass through the same
point they are said to be concurrent.
Prop. 4.-The bisectors of the three sides of a triangle
are concurrent. A
Let BE, CD, the bisectors of
AC, AB, intersect in 0; the
Prop. will be proved by showing that AO produced bisects D /
BC. Through B draw BG 1I
to CD, meeting AO produced
in G; join CG. Then, be- B
cause DO bisects AB, and is \
[1 to BG, it bisects AG (2) in
O. Again, because OE bisects G
the sides AG, AC, of the A
AGC, it is 11 to GC (3). Hence the figure OBGC is
a =, and the diagonals bisect each other (1);.~. BC is
bisected in F.
Cor.-The bisectors of the sides of a L divide each
other in the ratio of 2: 1.
Because AO = OG and OG = 20F, AO = 20F.
Prop. 5.-The middle points E, F, G, I of the sides AC,
BC, AD, BD of two triangles ABC, ABD, on the same base
AB, are the angular points o
of a fparallelogram, whose
area is equal to half sum
or half difference of the areas
of the triangles, according as /
they are on opposite sides, A ~.
or on the same side of the \
common base. G
Dem. 1. Let the As
be on opposite sides. The
B2