CONIC SECTIONS, ANALYTICAL. "1 V* 



and the fourth is the pole of the given straight line ; prove that 

 four conies will nu-c-t the given straight line in the same 

 two points, and that these points are the points of contact of the 

 two conies through the four points touching the line. 



876. Four straight lines and a point being given, four conic? 

 are described, with respect to each of which three of the four lines 

 form a self-conjugate triangle, and the fourth is the polar of the 

 given point ; prove that all four will have two common tangents 

 through the given point, and these tangents are tangents to the 

 two conies through the point touching the four lines. If two 

 tangents be drawn through the point to -any conic touching the 

 four lines, these will form a harmonic pencil with the two common 

 tangents. 



877. Four tangents are drawn to a conic, and from a point T 

 on one of the diagonals of the quadrilateral formed by them two 

 other tangents are drawn ; prove that the points of contact of 

 these tangents lie on the conic passing through T, ami through 

 the points of intersection of the four tangents which do not lie on 

 the diagonal through T. 



878. CA, CB are two tangents to a conic, any point /' in A II 

 is joined to the point in which its polar meets a fixed straight 



; n>ve that the envelope of the joining line is a conic touch- 

 ing the sides of the triangle ABC and the fixed straight line. 



879. ABC is a triangle whose sides are im-t by a straight 

 A'. />', (7; the straight line whu-h joins A to the point 



(Bff 1 CC f ) meets BC in a;*and 6, c are .similarly d. In-mined. 

 Four conies are drawn touching the sides of the triangle ABC, and 

 meeting A'B'C in the same two points; prove that the other 

 on chord of any two of these conies passes through either 

 a, 6, or c; that these six common chords intersect by threes in 

 four points; and that these fur points are the poles of A'B'C 

 with respect to the four conies whi.-h intersect ARC? in tin 

 before-mentioned two points and touch the sides of tl 

 angle abc. 



