A'Q; and through the conjugate points B, B' draw parallel

straight lines meeting the former in P, Q respectively. Then, if QP, A'A meet in O,

Hence the centre O and the rectangle are determined.

19. The following are examples of two general methods by which it may be proved that a system of points is in involution.

Ex. 1. Let any number of circles be described passing through the same two points A, A', and let any straight line cut these circles in the pairs of points P, P'; Q, Q'; Let the two straight lines intersect in O.

....

since each of these rectangles is equal to OA.OA'. [Euc. III.,36. Hence P, P'; Q, Q'; ... are conjugate points of a system in involution, O being the centre.

Ex. 2. Through 0, the intersection of the interior diagonals of the complete quadrilateral in fig., p. 183, draw any straight line, cutting the third diagonal in O'. Let this straight line also cut the sides DA, CB in M, M', and the sides AB, DC in N, N'. Then will M, M'; N, N' be

conjugate points in a system in involution which has O, O' for foci.

For, since the pencil is harmonic, Art. 9, the range O', M, O, M' is harmonic. Similarly the range O', N, O, N' is harmonic; which proves the proposition.

EXAMPLES.

1. If a pencil cut two transversals, neither of which is parallel to one of its rays, in the points A, B, C, D; A', B', C', D', prove by a direct method that

{A'B'C'D'} = {ABCD}.

2. If a straight line passing through a fixed point cut two fixed straight lines, the straight lines which join the points of section to two fixed points intersect on a fixed conic.

3. A straight line having one extremity on a fixed straight line moves so as to subtend constant angles at two fixed points. Prove that its other extremity traces out a conic section.

4. The sides of a triangle pass through fixed points and two of its vertices move on fixed straight lines: determine the locus of the third.

5. Given three points of a harmonic range, show how to determine the fourth by a geometrical construction.

6. If an ellipse be inscribed in a triangle and one focus move along a fixed straight line, the locus of the other focus will be a conic passing through the angular points of the triangle.

7. If the sides of a triangle pass through fixed points lying in a straight line, and if two of its vertices lie on given straight lines, the locus of the third will be a pair of straight lines passing through the intersection of the given straight lines.

8. If two triangles circumscribe a conic their angular points will lie on another conic.

9. Two triangles are constructed, such that two sides of each are tangents to a conic, the chords of contact being the third sides, prove that the six angular points lie on a conic.

10. Assuming that the pencil which joins four fixed points on a conic to a variable point on the curve is constant, prove that, if from any point on a conic pairs of perpendiculars be drawn to the opposite sides of a given inscribed quadrilateral, the rectangle contained by one pair varies at that contained by the other.

11. The square of the perpendicular from a point on a conic upon any chord varies as the rectangle contained by the perpendiculars from that point upon the tangents at the extremities of the chord.

12. Hence prove that a chord of a conic is divided harmonically by any point in its length and the point in which it intersects the chord of contact of tangents through that point.

13. If normals be drawn to a parabola from any point on the curve, the chord joining their extremities will meet the axis in a fixed point.

14. PG is a chord of an ellipse normal at P, and QR any chord which subtends a right angle at P. Prove that the opposite sides of the quadrilateral PQGR intersect on a fixed straight line.

15. If A, B, C; A', B', C' be fixed points on two straight lines, and {A'B'C'D'}={ABCD}, determine the envelope of DD'.

16. Find the envelope of the base of a triangle inscribed in a conic, two of its sides passing through fixed points.

17. Three pairs of conjugate diameters of a conic form a pencil in involution.

18. A straight line is cut in involution by the sides and diagonals of a quadrilateral.

19. Straight lines drawn from any point to the extremities of the diagonals of a complete quadrilateral form a pencil in involution.

20. A straight line is cut in involution by any conic and the sides of an inscribed quadrilateral.

CHAPTER XIV.

POLES AND POLARS.

The Polar of a fixed point with respect to a conic is the straight line which is the locus of intersection of tangents at the extremities of a chord which passes through the fixed point. See p. 87.

Conversely, if pairs of tangents be drawn to a conic from points on a fixed straight line, the fixed point through which their chords of contact pass is said to be the Pole of the fixed straight line.

A triangle is said to be self-conjugate with respect to a conic when each vertex is the pole of the opposite side.

1. If a chord of a conic pass through a fixed point, the tangents at its extremities will intersect on a fixed straight line. Let LN be the chord of contact of tangents drawn from