CONIC SECTIONS, ANALYTICAL. 183 



fuur such points corresponding to the triangles formed by four 

 given straight lines will lie on one straight line. 



862. The locus of a point, such that the tangents drawn 

 from it to a given conic are harmonically conjugate to the straight 

 lines joining it to two given points is a conic passing through 

 the two given points, and through the points of contact of the 

 tangents drawn from the two points to the given conic. This locus 

 reduces to a straight line, if the line joining the two points touches 

 the given conic. 



863. Two conies touch each other at 0, any straight line 

 through meets them in P, Q ; prove that the tangents at P t Q 

 intersect ill the straight line joining the two other points of 

 intersection of the conic. 



864. Four tangents A, B, C, D are drawn to a conic, the 

 line joining the points of contact of A, B meets C, D respectively 

 in P, Q ', prove that if a conic be described touching the four 

 lines, and P be its point of contact with C, Q will be its point of 

 contact with D. 



8C5. Two conies S, S' intersect in A, B, C, D, and the pole 

 of AH with respect to S is the pole of CD with respect to/S"; 

 prove that the i>ole of CD with respect to S is the pole of Alt 

 with respect to b". 



866. A quadrilateral can be projected into a rhombus on any 

 plane parallel to one of its diagonals, the vertex being any point 

 on a circle in a certain parallel plane. 



867. If ABC be a triangle circumscribing a conic, A' the 

 of contact of /f<7, 1) tin- point where AA' again meets the 



in<l /'/' be any tangent meeting the tangent at D in T \ 

 j..-ncil T{ABCT\ will l>o harmonic. 



868. ABC in a triangle circumscribing a conic, 77', TQ two 

 tangents, a conic is described about TPQBC, and is the 



with reapect to it ; prove that A {OBCT} is harm*. 



