170 BOOK OF MATHEMATICAL PROBLEMS. 



809. Through a given point are drawn chords PP' 1 QQ' to a 

 given conic, so as to touch any the same confocal conic ; prove 

 that the points of intersection of PQ, PQ, and of PQ', P'Q are 

 fixed. 



810. If A, B be two fixed points, P, Q any two points on the 

 same straight line, such that the range {APQB} is harmonic, and 

 a circle be described on PQ as diameter; all such circles will 

 have a common radical axis, and will cut orthogonally any circle 

 passing through A, B. 



811. If two triangles be formed, each by two tangents to a 

 conic and the chord of contact, the six angular points lie on 

 a conic. 



812. Four chords of a conic are drawn through a point, and 

 two other conies are drawn through the point, and each through 

 four extremities of the chords, respectively opposite ; prove that 

 these conies will have contact at the point. 



813. Through a given point is drawn any straight line 

 meeting a given conic in Q, Q', and P is taken on this line, so 

 that the range [OQQ'P] is constant ; prove that the locus of P is 

 an arc of a conic having double contact with the former. 



814. Given two points A, B of a conic, the envelope of a 

 chord PQ, such that the pencil {APQB} on the conic is equal to a 

 given quantity, is a conic touching the former conic at A, B. 



815. Through a fixed point is drawn any straight line meet- 

 ing two fixed straight lines in Q, R respectively ; E t F are two 

 other fixed points, QE, RF meet in P; prove that the locus 

 of P is a conic passing through E, F t and the point of intersec- 

 tion of the two fixed straight lines. 



816. If A, B, C be three points on a conic, and P, P' any 

 two other points, such that the pencils {PABC}, {FCBA} at any 

 point on the conic are equal j PP', CA, and the tangent at B, will 

 meet in a point. 



