166 BOOK OF MATHEMATICAL PROBLEMS. 



799. The straight lines 



xa + yfi + zy = 0, x'a 4- yft 4- z'y = 0, 



will be conjugate with respect to all parabolas inscribed in the 

 triangle of reference, if 



yz' -f yz = zx' + z'x = xy + x'y : 



and with respect to all conies touching the four straight lines 

 la * m/3 ny - 0, 



xx 1 _yy f __**' 



it ~ij- = : = . . 



I m n 



800. A rectangular hyperbola is inscribed in the triangle 

 ABC ; prove that the locus of the pole of the straight line which 

 bisects the sides AB, AC is the circle 



a' (a 1 + 6* + (?) + (/? + 2a) (a 1 + 6 s - c') + (/ + 2ya) (a 1 - 6 + c f ) = 0. 



801. Four conies are described with respect to any one of 

 which three of the four straight lines lam/3ny = Q form a 

 sell-conjugate triangle, and the fourth is the polar of a fixed 

 point (a', /J 7 , /) ; prove that the four will have two common 

 tangents meeting in (a', /?, y 7 ), their equation being 



^ 



Py'-Py 



802. A triangle circumscribes the conic 



la* + w/3* + ny* = 0, 

 and two of its angular points lie on the conic 



prove that the locus of the third angular point is the conic 



, -: ,.+...+...-0, 



( m + !^ _ V 

 \m n l) 



