Plane trigonometry, by S.L. Loney.

230 TRIGONOMETRY.
The relation found above is therefore true for all
triangles.
Hence, in all three cases, we have
R = a b = C (Art. 163).
2 sinA 2 sin B 2 sin C
201. In Art. 169 we have shewn that
2S
sin A 2 2S
bc s ) ) be
where S is the area of the triangle.
Substituting this value of sin A in (1), we have
abc
4S'
giving the radius of the circumcircle in terms of the sides.
202. To find the value of r, the radius of the incircle
of the triangle ABC.
Bisect the two angles B and C by the two lines BI
and CI meeting in I.
By Euc. III. 4, I is the A
centre of the incircle. Join
IA, and draw ID, IE and E
IF perpendicular to the
three sides. /..I" i
Then ID=IE= IF=r. B D C
We have
area of A IBC =.ID. BC = r. a,
area of A ICA = 2IE. CA = r. b,
and area of A IAB = iIF. AB = r. c.