654. The normal at a point P is produced to O so that PO is bisected by the axis: prove that any chord through O subtends a right angle at P; and that the circle on PO as diameter will have double contact with the parabola.

655. From a fixed point 0 is let fall 0Q perpendicular on the diameter through a point P of a parabola: prove that the perpendicular from on the tangent at P will pass through a fixed point, which remains the same for all equal parabolas on a common axis.

656. A circle is drawn through two fixed points R, S, and meets a fixed straight line through R again in P: prove that the tangent at P will touch a fixed parabola whose focus is S.

657. Two fixed straight lines intersect in 0: prove that any circle through and through another fixed point S meets the two fixed lines again in points such that the chord joining them touches a fixed parabola whose focus is S.

658. The perpendicular AZ on the tangent at P meets the parabola again in Q prove that the rectangle ZA, AQ is equal to the square on the semi latus rectum and that PQ passes through the centre of curvature at 4.

659. Two parabolas have a common focus and axes at right angles : prove that the directrix of either passes through the point of contact of their common tangent with the other.

660. Through any point P on a parabola is drawn PK at right angles to AP to meet the axis in K: prove that AK is equal to the focal chord parallel to AP. Explain the result when P coincides with A.

661. A circle on a double ordinate to the axis PP' meets the parabola again in Q, Q: prove that the latus rectum of the parabola which touches PQ, PQ', P'Q, P'Q' is double that of the former, and its focus is the centre of the circle.

662. Three points A, B, C are taken on a parabola, and tangents drawn at them forming a triangle A'B'C'; a, b, c are the centres of the circles BC'A', CAB, ABC': prove that the circle through a, b, c will pass through the focus.

663. Two points are taken on a parabola, such that the sum of the parts of the normals intercepted between the points and the axis is equal to the part of the axis intercepted between the normals: prove that the difference of the normals is equal to the latus rectum.

664. The perpendicular SY being drawn to any tangent, a straight line is drawn through Y parallel to the axis to meet in Q the straight line through S parallel to the tangent: prove that the locus of Q is a parabola.

665. If I be the foot of the directrix, SY perpendicular from the focus on a chord PP', and a circle with centre S and radius equal to XY meet the chord in QQ': prove that PP', QQ' subtend equal angles at S.

666. A given straight line meets one of a series of coaxial circles in A, B: prove that the parabola which touches the given straight line, the tangents to the circle at A, B, and the common radical axis will have another fixed tangent.

[If K be a point circle of the system, L the intersection of the given straight line with the radical axis and KO drawn at right angles to KL to meet the radical axis in 0, the fixed tangent is the straight line through O perpendicular to the given straight line.]

667. Two tangents TP, TQ are drawn to a parabola, OP, OQ are tangents to the circle TPQ: prove that TO will pass through the focus.

668. A triangle ABC is inscribed in a circle, Al' is a diameter, a parabola is described touching the sides of the triangle with its directrix passing through A' and S is its focus: prove that the tangents to the circle at B, C will intersect on SA'.

669. The normals at two points P, Q meet the axis in p, q and the chord PQ meets it in 0: prove that straight lines drawn through 0, p, q at right angles respectively to the three lines will meet in a point.

670. Normals at P, P' meet the axis in G, G', and straight lines at right angles to the normals from G, G' meet in Q: prove that

671. The tangent to a parabola at P meets the tangent at Q in T and meets SQ in R; also the tangent at Q meets the directrix in K: prove that PT, TR subtend equal or supplementary angles at K.

672. Two equal parabolas have a common focus and axes inclined at an angle of 120": prove that a tangent to either curve at a common point will meet the other in a point of contact of a common tangent.

673. The chord PR is normal at P, O is the centre of curvature at P and U the pole of PR: prove that OU will be perpendicular to SP.

674. From a fixed point O is drawn a straight line OP to any point P on a fixed straight line: prove that the straight lines drawn through Pequally inclined to PO and to the fixed straight line touch a fixed parabola.

675. A parabola whose focus lies on a fixed circle and whose directrix is given, always touches two fixed parabolas whose common focus is the given centre, and whose directrices are each at a distance from the given directrix equal to the given radius; and the tangents at the points of contact are at right angles.

676. The centre of curvature at P is 0, PO meets the axis in G and OL is drawn perpendicular to the axis to meet the diameter through P: prove that LG is parallel to the tangent at P.

677. The straight lines Aa, Bb, Ce are drawn perpendicular to the sides BC, CA, AB of a triangle ABC: prove that two parabolas can be drawn touching the sides of the triangles ABC, abe respectively, such that the tangent at the vertex of the former is the axis of the latter.

678. A right-angled triangle is described self-conjugate to a given parabola and with its hypotenuse in a given direction: prove that its vertex lies on a fixed straight line parallel to the axis of the parabola and its sides touch a fixed parabola.

679. Two equal parabolas have their axes in the same straight line and their vertices at a distance equal to the latus rectum; a chord of the outer touches the inner and on it as diameter is described a circle: prove that this will touch the outer parabola.

680. Tangents are drawn from a fixed point 0 to a series of confocal parabolas: prove that the corresponding normals envelope a fixed parabola whose directrix passes through O and is parallel to the axis of the system, and whose focus S" is such that OS is bisected by S.

681. A point O on the directrix is joined to the focus S and SO bisected in F; with focus F is described another parabola whose axis is the tangent at the vertex of the former and from 0 two tangents are drawn to the latter parabola: prove that the chord of contact and the corresponding normals all touch the given parabola.

682. Prove the following construction for inscribing in a parabola a triangle with its sides in given directions :-Draw tangents in the given directions touching at A, B, C, and chords AA', BB, CC' parallel to BC, CA, AB; A'B'C' will be the required triangle.

[The construction is not limited to the parabola, and a similar construction may be made for an inscribed polygon.]

683. Two fixed tangents are drawn to a parabola: prove that the centre of the nine points' circle of the triangle formed by these and any other tangent is a straight line.

684. At one extremity of a given finite straight line is drawn any circle touching the line, and from the other extremity is drawn a tangent to this circle: prove that the point of intersection of this tangent with the tangent parallel to the given line lies on a fixed parabola, and those with the tangents perpendicular to the given line on two fixed hyperbolas.

685. Two parabolas have a common focus and from any point on their common tangent are drawn other tangents to the two: prove that the distances of these from the focus are in a constant ratio.

686. Two tangents are drawn to a parabola equally inclined to a given straight line: prove that their point of intersection lies on a fixed straight line passing through the focus.

687. Two parabolas have a common focus S, parallel tangents drawn to them at P, Q meet their common tangent in P, Q': prove that the angles PSQ, PSQ' are each equal to the angle between the axes.

688. Two parabolas have parallel axes and two parallel tangents are drawn to them: prove that the straight line joining the points of contact passes through a fixed point.

[A general property of similar and similarly situate figures.]

689. On a tangent are taken two points equidistant from the focus: prove that the other tangents drawn from these points will intersect on the axis.

690. A circle is described on the latus rectum as diameter and a straight line through the focus meets the two curves in P, Q: prove that the tangents at P, Q will intersect either on the latus rectum or on a straight line parallel to the latus rectum and at a distance from it equal to the latus rectum.

691. A chord is drawn in a given direction and on it as diameter a circle is described: prove that the distance between the middle points of this chord and of the other common chord of the circle and parabola is of constant length.

692. On any chord as diameter is described a circle cutting the parabola again in two points: prove that the part of the axis of the parabola intercepted between the two common chords is equal to the latus rectum.

693. 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 of which this chord is a diameter touches the parabola of which this is a chord.

694. A parabola is described having its focus on the arc, its axis parallel to the axis, and touching the directrix, of a given parabola : prove that the two curves will touch each other.

695. Circles are described having for diameters a series of parallel chords of a parabola: prove that they will all touch another parabola related to the given one in the manner described in the last question.

696. A circle is described having double contact with a parabola and a chord QQ' of the parabola touches the circle in P: prove that QP, Q'P are respectively equal to the distances of Q, Q' from the common chord.

697. The locus of the centre of the circle circumscribing the triangle formed by two fixed tangents to a parabola and any other tangent is a straight line.

698. The locus of the focus of a parabola touching two fixed straight lines one of them at a given point is a circle.

699. Two equal parabolas A, B have a common vertex and axes opposite prove that the locus of the poles with respect to A of tangents to B is A.

700. Three common tangents PP', QQ, RR' are drawn to two parabolas and PQ, P'Q' intersect in L: prove that LR, LR' are parallel to the axes. Also prove that if PP' bisect QQ' it will also bisect RR', ... PP' will be divided harmonically by QQ', RR.

701. Two equal parabolas have a common focus and axes opposite; two circles are described touching each other, each with its centre on one parabola and touching the tangent at the vertex of that parabola: prove that the rectangle under their radii is constant whether the contact be internal or external, but in the former case is four times as great as in the latter.

702. Two equal parabolas have their axes parallel and opposite, and one passes through the centre of curvature at the vertex of the other: prove that this relation is reciprocal and that the parabolas cut at right angles.

703. From the ends of a chord PP' are let fall perpendiculars PM, P'M' on the tangent at the vertex: prove that the circle on PP' as diameter and the circle of curvature at the vertex have PP' for radical axis.

[The analytical proof of this is instantaneous.]

704. A parabola touches the sides of a triangle ABC in A', B', C', B'C' meets BC in P, another parabola is drawn touching the sides and P is its point of contact with BC: prove that its axis is parallel to B'C'.

705. The directrix and one point being given, prove that the parabola will touch a fixed parabola to which the given straight line is tangent at the vertex.

706. The locus of the focus of a parabola which touches a given parabola and has a given directrix parallel to that of the given parabola is a circle.

707. A triangle is self-conjugate to a parabola, prove that the straight lines joining the mid points of its sides touch the parabola; and that the straight line joining any angular point of the triangle to the point of contact of the corresponding tangent will be parallel to the

axis.

708. Four tangents are drawn to a parabola: prove that the three circles whose diameters are the diagonals of the quadrilateral will have the directrix as common radical axis.

709. A circle is drawn meeting a parabola in four points and tangents drawn to the parabola at these points: prove that the axis of the parabola will bisect the diagonals of the quadrilateral so formed.

710. The tangents at P, Q meet in T, and O is the centre of the circle TPQ: prove that OT subtends a right angle at S and that the circle OPQ passes through S.

711. Three parallels are drawn through A, B, C to meet the opposite sides of the triangle ABC in A', B', C": prove that a parabola can be drawn through A'B'C' and the middle points of the sides, and that its axis will be in the same direction as the three parallels.