Plane trigonometry, by S.L. Loney.

RADII OF THE ESCRIBED CIRCLES.

235

B- C. B C B. C\. acos - cos = r sin - cos - + cos - sin
=r1 sin ( + 2) = rsin (90~ -2 = r cos 2.
B C
cos cos
cos 2
r1=a A
COS 2
A A
Cor. Since a = 2R sin A = 4R sin A cos
2 2i
A B U
we have r = 4R sin cos cos
2 2 2
EXAMPLES. XXXVI.
1. In a triangle whose sides are 18, 24, and 30 inches respectively,
prove that the circumradius, the inradius, and the radii of the three
escribed circles are respectively 15, 6, 12, 18, and 36 inches.
2. The sides of a triangle are 13, 14, and 15 feet; prove that
(1) R=8s ft., (2) r=4 ft., (3) rI=10, ft.,
(4) r2= 12 ft.,and (5) r3 =14 ft.
3. In a triangle ABC if a=13, b=4, and cos = -, find
13'
R, r7, '1, '2, and 1-3
4. In the ambiguous case of the solution of triangles prove that the
circumcircles of the two triangles are equal.
Prove that
5. r1+r2+r3-r=4R. 6.?1?2+J2r23+23rl=s2
A B C
7. r1r2 r3 cot2 cot2 cot2. 8. rrl2?r9' S2.
9. — + - - -0. 10. S = 2R2sin A sin B sin C.
rl r2 7'3 r