

PLANE TRIGONOMETRY. 71 



IV. Properties of Triangles. 



In these questions a, b, c denote the sides, A, B, C the re- 

 spectively opposite angles of any triangle; 7? the radius of the 

 circumscribed circle, r, r,, r s , r a the radii of the inscribed circle 

 and of the escribed circles respectively opposite A, B y C. 



330. If 0, <, \f/ be angles given by the equations 



a b c 



cos = -? , cos0 = . cosuf = r : 



b + c c + a a + b 



then will tan' | + tan' + tan 9 = 1 ; 



J J J 



0. <x ^ A >. B L C 



and tan tan J tan = * tan ^ tan -^ tan ^ . 



* v J SI 2 J 



331. If sin ,4, sin J?, sin C be in harmonical progression, so 

 also are 1 cos A, 1 cos B, 1 cos C. 



332. Prove that 



wn A sin (A - B) sin (A - C) + sin J5 sm(J5- C) sin (#-^) 



+ sin C sin ((7- 4) sin (C- B) = sin A sin 5 sin (7 



- sin 24 sin 25 sin 2(7. 



333. From the three relations between the sides and anglea 

 i in the form 



deduce the equations 



assuming that each angle is < 180. 



