PL ANT, TRIGONOMETRY. 77 



If p be the radius of this latter circle, and 8 the distance of 

 6 from the centre of the circumscribed circle, 



2G4. The line joining the centres of the circumscribed and 

 inscribed circles will subtend a right angle at the centre of per- 

 pendiculars, if 



1 + (1 - 2 cos A) (1 -2 cos B) (1 - 2 cosC) = ScosA coaJB cos C. 



36,">. If P be a point within a triangle at which the sides 

 subtend angles A + a, B + /?, C + y, respectively, 



J . 



sin a sin p sin y 



366. Any point P is taken within the triangle ABC, and tin- 

 angles BPC f CPA, APB are A', B', C' respectively; prove that 



&BPC(cot A-cotA') = & CPA (cot B - cot B) 



= A APB (cot C- cot C"). 



V. If ijhts and Distances. Polygons. 



7. At a point ^ are measured the angle a subtended 1 >y 



two objects P, $ in the same horizontal plane as A and tl it- 



distances a, b at right angles to AP, AQ respectively to points at 



which PQ subtends the same angle a; find the distance between 



1 Q. 



-\ An object is observed at three points A, B, C lying in :. 



ni.il lino which passes directly underneath the object; the 

 angular elevation at B is twice and at C is three times that at A ; 

 also AB = a, BC = b', prove that iho height of the object is 



If the angle of elevation at A be tan~ l J, a : b ::!,". 



