Plane trigonometry, by S.L. Loney.

236

TRIGONOMETRY.

[Exsc XXXVI.]

11.. 4R sin A sin B sin C= a cos A + b cos B + c cos C.
A B C I A
12. S =4R? cos cos cos. 13. r-1 =tan2.
2 2 2s r223 2
14. r1 (s - a) =-2 (s - b) = r?3 (s - c) =rs = S.
15. a (rr1+ r2r3)=6b (rr.2+r3rl)=c ('3 + r1?2).
i1 1 1 1 a2 + b2+c2
167. c+ — S + - - 2
s rl 7.22.32.
17. r-l cot 2 =S. 18. (r - r)(r - ) (r73-r)=4R.2.
C C
19. (r1 +r2) tan 2 = (r3 - ') cot =c.
1 1 1 1 r r9 1 1
20. + _+ —= 21. + + - 1
C ca 21r 2 be ca ab r 2R
22. 2 + r2 + rr22 + r32 =16R - a2- b2 -c2.
208. Orthocentre and pedal triangle of any
triangle.
Let ABC be any triangle and let AK, BL, and CM be
the perpendiculars from A, B, and C
upon the opposite sides of the tri- A
angle. It can be easily shewn, as M,.
in most editions of Euclid, that / -L
these three perpendiculars meet in'
a common point P. This point P B
is called the orthocentre of the
triangle. The triangle KLM, which is formed by joining
the feet of these perpendiculars, is called the pedal
triangle of ABC.
209. Distances of the orthocentre from the angular
points of the triangle.