164 BOOK OF MATHEMATICAL PROBLEMS. 



787. Given a point on a conic, and a triangle ABC self- 

 conjugate to the conic, AO, BO, CO meet the opposite sides in 

 three points, and the lines joining these two and two meet the 

 corresponding sides in A', K, G' ; prove that the intersections of 

 BB', CC' ', CC', A A'-, and AA', BE also lie on the conic. 



788. Any tangent to a conic meets the sides of a self-conju- 

 g-ito triangle in Z>, E, F; the line joining A to the intersection of 

 BE, CF meets BC in A' ; J?, C' are similarly determined ; provs 

 that B'C' t C'A', A'B' are also tangents to the conic. 



789. Tf la. + w/3 4- ny = be the equation of an asymptote 

 of a rectangular hyperbola self-conjugate to the triangle of 



reference, 



la 9 mih* nc* _ 



+ , + j = 0. 



m n nl l m 



790. A parabola is described touching the sides of a triangle 

 ABC, S \& the focus, and the axis meets the circle circumscribing 

 the triangle again in 0; prove that if with centre a rectangular 

 hyperbola be described, to which the triangle is self-conjugate, one 

 of its asymptotes will coincide with OS. 



791. One directrix of the conic 



la 3 + mfi* + ny* = Q 

 passes through A' } prove that 



1 _ cot 8 B cot 8 C t 

 I m n ' 



and that the second focus lies on the line joining the feet of the 

 perpendiculars from B, C on the opposite sides. 



792. The equations determining the foci of the conic 



la 9 + mfi + ny* = 



are 



sin" A \ & m 



sin 8 C { n 



' + F + ) 



n I) 



