182 BOOK OP MATHEMATICAL PROBLEMS. 



855. A conic touches the sides of a triangle ABC in D, E, F t 

 and AD, BE, CF meet in Sy three conies are described with S as 

 focus, osculating the former at D, E, F ', prove that these three 

 and the former will have one common tangent, which also touches 

 the conic having S as focus and touching the sides of the triangle 

 ABC. 



856. Given four straight lines, prove that two conies can be 

 constructed so that an assigned straight line of the four is its 

 directrix and the other three form a self-conjugate triangle : and 

 that, whichever straight line be taken for directrix, the corre- 

 sponding focus will be one of two fixed points. 



857. Through a fixed point are drawn two chords PP f t 

 QQ f to a given conic, such that the two lines bisecting the angles 

 at are also fixed ; prove that the straight lines PQ> P'Q, PQ\ 

 P'Q' all touch a fixed conic ; except when the two fixed straight 

 lines are conjugates to the given conic. 



858. OP, OQ ; 07* O'Q are tangents to a conic, and the 

 conic which touches the sides of the triangles OPQ, O'P'Q' is 

 drawn : prove that any tangent to the latter conic will be divided 

 harmonically by the former conic and the lines PQ, P'Q'. 



859. Two conies circumscribe the triangle ABC, any straight 

 line through A meets them again in P, Q ; and the tangents at 

 /*, Q meet JBC in P'Q' ; prove that the range {BPQ'C} is con- 

 stant. 



860. The equation of the polar reciprocal of the evolute of 

 the ellipse - t + TJ = 1 with respect to the centre is 



o f b a a -6 J 



861. If P, Q be two fixed points, and if on the side BC of a 

 triangle ABC be taken a point A', such that the pencil A' {APQB} 

 is harmonic; and R, C' be similarly taken on the sides CA, AB-, 

 the straight lines A A', Bff, CQ' will meet in a point j and the 



