24. Shew that the nine-point circle of a triangle touches the inscribed circle and each of the escribed circles.

25. The tangents to the nine-point circle at the points where it touches the inscribed and escribed circles form a quadrilateral, each diagonal of which passes through an angular point of the triangle, and the lines joining corresponding angular points of the original triangle and of the triangle formed by the diagonals are all parallel to the radical axis of the nine-point circle and the circumscribing circle.

26. The polars of the points A, B, C with respect to a conic are B'C', C'A', A'B' respectively; shew that AA', BB', CC' meet in a point.

27. If an equilateral hyperbola pass through the middle points of the sides of a triangle ABC and cuts the sides BC, CA, AB again in a, ß, y respectively, then Aa, Bß, Cy meet in a point on the circumscribed circle of the triangle ABC.

28. Shew that the locus of the intersection of the polars of all points in a given straight line with respect to two given conics is a conic circumscribing their common self-conjugate triangle.

29. Two conics have double contact; shew that the locus of the poles with respect to one conic of the tangents to the other is a conic which has double contact with both at their common points.

30. Two triangles are inscribed in a conic; shew that their six sides touch another conic.

31. Two triangles are self-polar with respect to a conic; shew that their six angular points are on a second conic, and that their six sides touch a third conic.

32. If one triangle can be described self-polar to a given conic and with its angular points on another given conic, an infinite number of triangles can be so described.

33. A system of similar conics have a common self-conjugate triangle; shew that their centres are on a curve of the 4th degree which passes through the circular points at infinity and of which the angular points of the triangle are double points.

34. If A, B, C, A', B', C' be six points such that AA', BB', CC' meet in a point, then will the six straight lines AB', AC', BC', BA', CA' and CB' touch a conic.

35. A conic is inscribed in a triangle and is such that the normals at the points of contact meet in a point; prove that the point of concurrence describes a cubic curve whose asymptotes are perpendicular to the sides of the triangle.

36. If P1, P2, Pa, P. be the lengths of the perpendiculars drawn from the vertices A, B, C, D of a quadrilateral circumscribed about a conic on any other tangent to the conic, shew that the ratio of P1P3 to P2P4 will be constant.

37. The polars with respect to any conic of the angular points A, B, C of a triangle meet the opposite sides in A', B', C'; shew that the circles on AA', BB', CC' as diameters have a common radical axis.

38. A parabola touches one side of a triangle in its middle point, and the other two sides produced; prove that the perpendiculars drawn from the angular points of the triangle upon any tangent to the parabola are in harmonical progression.

39. Shew that the tangential equation of the circumscribing circle is a √p+b √q+cr=0. Hence shew that the tangential equation of the nine-point circle is

a√(q + r) + b √(r + p) + c √(p + q).

40. The locus of the centre of a conic inscribed in a given triangle, and having the sum of the squares of its axis constant, is a circle.

41. The director circles of all conics inscribed in the same triangle are cut orthogonally by the circle to which the triangle of reference is self-polar.

42. The circles described on the diagonals of a complete quadrilateral are cut orthogonally by the circle round the triangle formed by the diagonals.

43. If three conics circumscribe the same quadrilateral, shew that a common tangent to any two is cut harmonically by the third.

44. If three conics are inscribed in the same quadrilateral the tangents to two of them at a common point and the tangents to the third from that point form a harmonic pencil.

45. The locus of a point the pairs of tangents from which to two given conics form a harmonic pencil is a conic on which lie the eight points in which the given conics touch their common tangents.

46. The locus of a point from which the tangents drawn to two equal circles form a harmonic pencil is a conic, which is an ellipse if the circles cut at an angle less than a right angle, and two parallel straight lines if they cut at right angles.

47. A triangle is circumscribed about one conic and two of its angular points are on a second conic; find the locus of the third angular point.

48. A triangle is inscribed in one conic and two of its sides touch a second conic; find the envelope of the third side.

49. The angular points of a triangle are on the sides of a given triangle, and two of its sides pass through fixed points; shew that the third side will envelope a conic.

50. From the angular points of the fundamental triangle pairs of tangents are drawn to (uvwu'v'w'(xyz)2 = 0, and each pair determine with the opposite sides a pair of points. Find the equation to the conic on which these six points lie, and shew that the conic

'x (v'w' — uu') + √y (w'u' − vv') + √z (u'v′ — ww') = 0 and the above two conics have a common inscribed quadrilateral.

CHAPTER XIV.

RECIPROCAL POLARS. PROJECTIONS.

300. If we have any figure consisting of any number of points and straight lines in a plane, and we take the polars of those points and the poles of the lines, with respect to a fixed conic C, we obtain another figure which is called the polar reciprocal of the former with respect to the auxiliary conic C.

When a point in one figure and a line in the reciprocal figure are pole and polar with respect to the auxiliary conic C, we shall say that they correspond to one another.

If in one figure we have a curve S the lines which correspond to the different points of S will all touch some curve S. Let the lines corresponding to the two points P, Q of S meet in T; then T is the pole of the line PQ with respect to C, that is the line PQ corresponds to the point T. Now, if the point Q move up to and ultimately coincide with P, the two corresponding tangents to S will also ultimately coincide with one another, and their point of intersection T will ultimately be on the curve S'. So that a tangent to the curve S corresponds to a point on the curve S', just as a tangent to S corresponds to a point on S. Hence we see that S is generated from S' exactly as S" is from S.

301. If any line L cut the curve S in any number of points P, Q, R...we shall have tangents to S' corresponding to the points P, Q, R..., and these tangents will all pass through a point, viz. through the pole of L with respect to the auxiliary conic. Hence as many tangents to S can be drawn through a point as there are points on S lying on a

straight line. That is to say the class [Art. 240] of S' is equal to the degree of S. Reciprocally the degree of S' is equal to the class of S.

In particular, if S be a conic it is of the second degree, and of the second class. Hence the reciprocal curve is of the second class, and of the second degree, and is therefore also a conic.

302.

To find the polar reciprocal of one conic with respect to another.

Let the equation of the auxiliary conic be

ax2 + By2+1=0...........................................(i);

and let the equation of the conic whose reciprocal is required be

ax2+by2+c+2fy + 2gx + 2hxy = 0 .......(ii).

The line lx+my+n = 0 will touch (ii) if

Al2+Bm2 + Cn2 +2Fmn + 2Gnl+2 Hlm = 0 (iii).

And, if the pole of lx+my+n=0 with respect to (i) be (x, y), its equation is the same as ax'x +ẞy'y +1 = 0. Therefore

ax'

=

m

By

Substitute, in (iii), and we have

/2

=

n

1

Aa +BB*ý* +C+2F®ý+2Gác +2 Haß ý =0. Hence the locus of the poles with respect to (i) of tangents to (ii) is the conic whose equation is

Aa2x2+BB'y2+ C + 2Fẞy + 2Gαx + 2Hαßxy = 0.

303. The method of Reciprocal Polars enables us to obtain from any given theorem concerning the positions of points and lines, another theorem in which straight lines take the place of points and points of straight lines. Before proceeding to give examples of such reciprocal theorems we will give some simple cases of correspondence.

Points in one figure correspond to straight lines in the reciprocal figure.

The line joining two points in one figure corresponds to the point of intersection of the corresponding lines in the other.