The pole of a fixed straight line, with respect to a conic inscribed in a given quadrilateral, lies on a fixed right line.

If two triangles be polar reciprocals with respect to any conic, the intersections of their corresponding sides lie on a straight line.

The polar of a fixed point, with respect to a conic circumscribing a given quadrilateral, passes through a fixed point.

If two triangles be polar reciprocals with respect to any conic, the straight lines which join their corresponding vertices meet in a point.

EXAMPLES.

1. OA, OB are tangents to a conic and CD a chord which subtends a right angle at A. Prove that, if AC bisect the angle OAB, then CD passes through 0.

2. Apply the method of the present chapter to draw tangents from a given point to a conic with the help of the ruler only.

Compare Ex. 24, p. 22.

3. Four straight lines, drawn from the same point, meet a conic in A, B, C, D; A', B', C', D'. Prove that

P{ABCD} = P{A'B'C'D'},

where P is any point on the curve.

4. AB, CD are parallel chords of a conic, and DE a chord which bisects AB. Prove that the tangents at C, E intersect on AB.

5. If the tangents to a parabola and their chords of contact be produced, two and two, show that each tangent is cut harmonically.

6. A quadrilateral being inscribed in a circle, a triangle is formed by joining the points of intersection of its diagonals and opposite sides. Prove that the perpendiculars from the angular points of this triangle upon the opposite sides pass through the centre of the circle.

7. A chord of a conic which subtends a right angle at a fixed point on the curve passes through a fixed point on the normal.

8. The anharmonic ratio of the pencil formed by any four diameters of a conic is equal to that of the pencil formed by the conjugate diameters.

9. Hence deduce that the anharmonic ratio of any range is equal to that of the pencil formed by the polars of the points which constitute the range.

10. Any number of points which lie on a straight line are in involution with the points in which their polars cut the straight line. Hence deduce the result of the last example.

11. The sum of a pair of supplemental chords equally inclined to the normal at their point of intersection is equal to the diameter of the circle which is the locus of intersection of tangents at right angles.

12. PQ is a chord of a circle whose pole lies on RS. The chord QT being parallel to RS, prove that PT bisects RS.

13. The envelope of the polar of a point on the circumference of a circle, with respect to a circle whose centre is S, is a conic having S for focus.

14. Prove that the eccentricity of the conic varies directly as the distance between the centres of the circles, and inversely as the radius of the former. Hence determine in what cases the conic will be an ellipse, a hyperbola, or a parabola respectively.

15. OA, OB are tangents to a conic; C any point on the Prove that if P, Q be points in AC, BC, such that OPQ is a straight line, then BP, AQ intersect on the curve.

curve.

16. A conic is inscribed in a triangle ABC, the points of contact being A', B', C'. If P be any point on B'C', the straight line joining the points in which BP, CP meet AC, AB respectively touches the conic.

17. If two triangles be so related that the sides of each are the polars of the vertices of the other, their six angular points will lie on a conic.

18. If two triangles be self-conjugate with respect to the same conic, their angular points will lie on a conic.

19. If two triangles be polar reciprocals with respect to a conic, the straight lines joining their corresponding vertices meet in a point, and the intersections of corresponding sides lie in a straight line.

20. Pp, Qq are chords of a conic parallel to the tangents QT, PT respectively. Prove that the tangents at p, q intersect on the diameter through T.

CHAPTER XV.

PROJECTION.

If all points of a plane figure be joined to a point not in the same plane, the joining lines form a Cone of which the fixed point is vertex, and the figure in which this cone is cut by any plane is said to be the Conical Projection, or simply the Projection, of the original figure.

The plane of the original figure will be called the Primitive Plane, and the cutting plane the Plane of Projection.

1. In the figure, let the large curve represent the projection of the small curve, V being the vertex. Then the two curves are so related that a straight line drawn from V to any point on the latter curve will pass through some point on the former. The second of these points is the projection of the first.

2. The projection of a straight line, as BO, is determined by the intersection of the plane BOV, drawn through that line and the vertex, with the plane of projection. Thus the projection of B() is the straight line BO'.

It is evident that if a straight line pass through a fixed point, its projection will pass through a fixed point, and if the locus of a point be a straight line, that of its projection will be a straight line.

In what follows, the primitive plane and the plane of projection are supposed to be fixed. The vertex has, in each case, to be determined.

3. To project an angle into any given angle.

Let AOB be any angle in the primitive plane; a, b points in AO, BO respectively. On ab describe a segment of a circle

(in a plane parallel to the plane of projection) containing an angle equal to that into which it is required to project AOB, and let V be any point on the segment.

A

V

α

B

Take V for vertex and let the plane of projection intersect the primitive plane in the straight line AB. Let O' be the projection of O. Then Va O'A is a plane. [Euc. XI., 2.

Now, since the parallel planes Vab, AO'B are intersected by the plane Va O'A in the straight lines Va, AO' respectively, therefore Va and AO' are parallel (Euc. XI., 16). Similarly Vb, BO' are parallel. Hence AO'B=aVb. [Euc. XI., 10. But AO' is the projection of AO, and BO' that of BO. Hence AO'B is the projection of AOB, and it is equal to a Vb or to the given angle.