EXAMPLES.

1. In the triangle ABC prove (1)

a+b:c=cos } (A – B): sin C, and (2)

a – b:c=sin ] (A – B): cos } C.

2. If AD bisects the angle A of the triangle ABC, prove

BD:DC = sin C:sin B.

3. If AD' bisects the external vertical angle A, prove

BD': CD' sin C:sin B.

COS A

- 2 sin)

А

.: 2

2 sin^

=

99. To express the Sine, the Cosine, and the Tangent of Half an Angle of a Triangle in Terms of the Sides. I. By Art. 96 we have 62 +c? — a? =1

(Art. 49) 2 bc

2
1, +6" -a?

2 bc
a' - (b-c)

2 bc
(a + b — c)(a − b + c).

2 bc Let

a+b+c=2 s; then

a+b-c=2(8-c), and a-b+c=2(8-0).

,- 2.
A
2

2

2 sin? 4 = 2 (8 —c)2(8 – 1).

(3)

II.

(Art. 49)

.: 2 cos2A

1+

sin

|(s – a)(s — b)
2
V

ab

A cos A = 2 cos?

os? A 1.

2

12 + c -a
2

2bc
(b + c)? – a?

2 bc
(a + b + c)(b + c-a)

2 bc
28.2 (s-a).

2 bc

=

S

C
tan
(s – a)(8 — b).

(9) S (8-0) Since any angle of a triangle is < 180°, the half angle is < 90°; therefore the positive sign must be given to the radicals which occur in this article.

COS

100. To express the Sine of an Angle in Terms of the Sides.

A A
sin A=
2 sin

(Art. 49)
2
b) ($

-c) 8 (8 - a).
2
bc

bc

(Art. 99) 2 ... sin A:

V8 (8 - a)(8 - ) (8 — c). bc

S

S

2
sin C Vs(s – a)(s—b)(s —c).

ab

1 Cor. sin A = V26%c" + 2 c?a? + 2 aob? – a'- 14-04,

2 bc and similar expressions for sin B, sin C.

EXAMPLES.

In any triangle ABC prove the following statements : 1. a (b cos C-c cos B)=b - c. 2. (b + c) cos A+(c + a) cosB +(a + b) cosC=a+b+c.

sin A +2 sin B sin C 3.

a +26
sin? A
- m sin’B

sinac
4.
a? - mb2

ca

с

7. a sin (B-C) + b sin (C – A) +csin (A – B)=0.

9. tan 4 A:tan B=($-)(--C).

(3) Given the three sides.

2 sin A Vs(s – a) (s — b)(8 —c) (Art. 100)

bc Substituting in

S= { bc sin A,

we get

S=Vs(s – a)(s – b)(s – c).

S

102. Inscribed Circle. — To find the radius of the inscribed circle of a triangle.

C Let ABC be a triangle, O the

F centre of the inscribed circle, and

E go its radius. Draw radii to the points of contact D, E, F; and join OA, OB, OC. Then

A

C D

ha

с

103. Circumscribed Circle. To find the radius of the circumscribed circle of a triangle in terms of the sides of the triangle. Let O be the centre of the circle A

B described about the triangle ABC, and R its radius.

Through O draw the diameter CD and join BD.

Then Z BDC =Ż BAC=LA.

.. R

abc

(2)