7G BOOK OF MATHEMATICAL PROBLEMS. 



358. If 0, o be the centres of the circumscribed and inscribed 

 circles, and L the centre of perpendiculars 



and if Oj be the centre of the escribed circle opposite A, 



3.39. If the centre of the inscribed circle, or of one of the 

 escribed circles, be equidistant from the centre of the circum- 

 scribed circle and from the centre of perpendiculars, one angle of 

 tin- triangle must be equal to 60. 



3GO. The angle at which the circumscribed circle of a tri- 

 angle intersects the escribed circle opposite A is 



! 1 + cos A - cos B - cos C 







If a, /?, y be the three such angles, 

 (cos ft + cos y) (cos y + cos a) (cos a + cos ft) 



= 2 (cos a 4- cos ft + cos y I) 2 . 



361. If P be any point on the circumscribed circle, 



PA sin A + PB sinB + PC sin C = 0, 



a certain convention being made with respect to sign : also 

 PA a sin 2A + PB'ain 2B + PC" sin 20 = 4 A ABC. 



362. If P be any point in the plane of the triangle, and 

 the centre of the circumscribed circle, 



PA 9 sin 2A + PB 9 sin 2B + PC" sin 2(7 



= 2 A ABC + WP a sin A sin B sin C. 



363. If P be any point on the inscribed circle, 



PA 9 sin A + PB a sin B + PC* sin C 



is constant; and if P be any point on the circle with respect to 

 which the triangle is self-conjugate, 



PA' i&nA + PB a ia,nB + PC a ten C =2 A ABC. 



