COXIC SECTIONS, GEOMETRICAL. !>1 



436. Two equal parabolas have a common focus and axes 

 opposite; two circles are described touching each other, each with 



utre on one parabola and touching the tangent at the vertex 

 of that parabola: prove that the rectangle nnder their radii is 

 constant whether the circles touch internally or externally, but in 

 the former case is four times as great as in the latter. 



437. Two equal parabolas are placed with their axes in the 

 same straight line and their vertices at a distance equal to the 

 latus rectum; a tangent drawn to one meets the other in two 

 points: prove that the circle on this chord as diameter will touch 

 the parabola of which this is the chord. 



438. Two equal parabolas have their axes parallel and oppo- 

 site, and one passes through the centre of curvature at the vertex 

 of the other; prove that this relation is reciprocal and that the 

 parabolas intersect at right angles. 



439. PP f is any chord of a parabola, FM, P f ^f' are drawn 

 perpendicular to the tangent at the vertex; prove that the circle 

 on J/J/' as diameter, and the circle of curvature at the vertex will 

 have Fl* for their radical axis. 



440. A parabola touches the sides of a triangle ABC in 

 J , //, C'j BC' meets BC in P, another parabola is drawn touching 



les and P is its point of contact with BC', prove that its 

 axis is paruIM to BC'. 



441. The directrix of a parabola and one point of the curve 

 being given ; prove that the parabola will touch a fixed parabola 

 to which the given straight lino is the tangent at the vertex, 



If a triangle be self-conjugate to a parabola, the lines 

 joining the middle point of its Hides will touch the parabola ; and 

 the linrs joining any angular point of the triangle to the point of 

 contact of the corresponding tangent will be parallel to tin- axis. 



443. If a complete quadrilateral be formed by t - Ul tangents 

 to a parabola, the common radical axis of the three circles on t !. 

 diagonals as diameters will be the directrix of the parabola, 



