Plane trigonometry, by S.L. Loney.

[Exs. XXXVII.] PROPERTIES OF TRIANGLES.

245

A
5. 12I3=a cosec 2. 6. II1.112. II3=16R2r.
B+C
7. 1,=3 =4R (r1+r3). 8. L 3 =12 2.
9. 112 + I223 = II2 + 13112 = II32 + 122.
10. Area of AIIIa3= 8R?2 cos cos cos a10. 2 22 2
11. 3II13 I2I112' 1a 13 III2
sin A sin B sin C
If I, 0, and P be respectively the incentre, circumcentre, and orthocentre, and G the centroid of the triangle ABC, prove that
12. IJ2=R2 (3 - 2 cos A - 2 cos B - 2 cos C).
13. P2= 2r?2 - 4R2 cos A cos B cos C.
14. OG2 =R2 (a + b2 + c2).
B-C. C-A. A-B
15. Area of AIOP= 22 sin, sin -2 sin A-2
2 2
2 B-C. C-A. A-B
16. Areaof AIPG- R2 sin sin - sin
3 2 2 2
17. Prove that the distance of the centre of the nine-point circle from
the angle A is /1l + 8 cos A sin B sin C.
18. DEE is the pedal triangle of ABC; prove that
(1) its area is 2S cos A cos B cos C,
(2) the radius of its circumcircle is -
and (3) the radius of its incircle is 2R cos A cos B cos C.
19. 010203 is the triangle formed by the centres of the escribed circles
of the triangle ABC; prove that
A B C
(1) its sides are 4R cos 2, 4R cos -, and 4R cos 2,
A r B 7r C
(2) its angles are - -2 and -
a2d (3) its area is 2Rs.
and (3) its area is 2/s.