114 BOOK OF MATHEMATICAL PROBLEMS. 



latus rectum is 27? sin sin (C - 0) sin (B + 0), JK being the radius 

 of the circumscribed circle of the triangle ; and if a parabola 

 touch the sides of the triangle, its latus rectum is 

 8R sin sin (G - 0) sin (B + 6). 



579. A triangle ABC is inscribed in a" parabola, and the 

 focus is the centre of perpendiculars of the triangle ; prove that 



(I -cos A) (1 - cos B) (1 - cosC) = 2 cos J. cosB cos (7; 



and that each side of the triangle will touch a fixed circle which 

 passes through the focus, and whose diameter is equal to the 

 latus rectum. 



580. Through a fixed point within a parabola is drawn any 

 chord PP', and the diameters through P and P' are drawn ; 

 prove that there are two fixed straight lines perpendicular to the 

 axis, the part of either of which intercepted between the dia- 

 meters subtends a right angle at 0. 



581. A triangle, self-conjugate to a given parabola, has one 

 angular point given ; prove that the circle circumscribing the 

 triangle passes through another fixed point Q, such that OQ is 

 parallel to the axis and bisected by the directrix. 



582. A triangle is inscribed in a parabola, its sides are at 

 distances x, y, z from the focus, and subtend angles 6, <, \j/ at the 

 focus ; prove that 



6 \!/ 



. ,. . , . , sin 6 + sin </> + sin \b + 2 tan s tan ~ tan ~ 

 sm sin < sin \p __ y r _ 222 



~tf~ ~Y~ ~* 2 ~^" 



Z being the latus rectum, and the angles being measured all in the 

 same direction so that their sum is 2?r. 



583. If a parabola touch the sides of a triangle ABC in 

 A', K, C', and be the point of intersection of A A', BB', CC', 

 then will 



OA OB PC 



sin BOG + am CO A + - ~ 



a certain convention being made as to sign. 



