Plane trigonometry, by S.L. Loney.

244 TRIGONOMETRY.
218. Bisectors of the angles.
If AD bisect the angle A and
divide the base into portions x and A
y, we have, by Euc. vi. 3,
x AB c
y AC b'. yx +y a 1 B D
cbb+o b- c.(1), D
"c b b+c - " b+c
giving x and y.
Also, if 8 be the length of AD and 0 the angle it
makes with BC, we have
AABD+ ACD = AABC.
1.. A 1 A 1
c s + sincsin sin bsinA,
bc sinA 2bc A
i.e. =.. cos -......... (2).
b+c. A b b+b c 2 (2
sin
A A A+2B
Also 8=180~-C- A -=A B- -—. (3)
2 2 2
We thus have the length of the bisector and its
inclination to BC.
EXAMPLES. XXXVII.
If I, I1, I2, and I1 be respectively the centres of the incircle and the
three escribed circles of a triangle ABC, prove that
A
1. AI=r cosec.
A B C
2. IA. IB. IC= abe tan - tan tan
2 2 2a
A A
3. AI,=r cosec2 4. II=asec -.
2' -i