9. The equation of the circumscribing conic, whose diameters parallel to the sides of the triangle of reference are T12r is

10.

b

ABC is a triangle inscribed in a conic, and the tangents to the conic at A, B, C are B'C', C'A', A'B' respectively; shew that AA', BB', and C'C' meet in a point. Shew also that, if D be the point of intersection of BC, B'C'; E the point of intersection of CA, C'A', and F the point of intersection of AB, A'B'; D, E, F will be a straight line.

11. Lines are drawn from the angular points A, B, C of a triangle through a point P to meet the opposite sides in A', B', C'. B'C' meets BC in K, C'A' meets CA in L, and A'B' meets AB in M. Shew that K, L, M are on a straight line. Shew also (i) that if P moves on a fixed straight line then KLM will touch a conic inscribed in the triangle ABC; (ii) that if P moves on a fixed conic circumscribing the triangle ABC, then KLM will pass through a fixed point; (iii) that if P moves on a fixed conic touching two sides of the triangle where they are met by the third, KLM will envelope a conic.

Lines drawn through the angular points A, B, C of a triangle and through a point O meet the opposite sides in A, B, C'; and those drawn through a point O' meet the opposite sides in A", B", C". If P be the point of intersection of B'C' and B'C", Q be the point of intersection of C'A', C"A", and R be the point of intersection of A'B', A"B"; shew that AP, BQ, CR will meet in some point Z. Shew also that, if 0, O' be any two points on a fixed conic through A, B, C, the point Z will be fixed.

13. The locus of the pole of a given straight line with respect to a system of conics through four given points is a conic which passes through the diagonal-points of the quadrangle formed by the given points.

14. The envelope of the polar of a given point with respect to a system of conics touching four given straight lines is a conic which touches the diagonals of the quadrilateral formed by the given lines.

15. Shew that the locus of the points of contact of tangents, drawn parallel to a fixed line, to the conics inscribed in a given quadrilateral, is a cubic; and notice any remarkable points, connected with the quadrilateral, through which the cubic passes.

16. An ellipse is inscribed within a triangle and has its centre at the centre of the circumscribing circle. Shew that its major and minor axes are R+d and R-d respectively, R being the radius of the circumscribing circle and d the distance between the centre and the orthocentre.

17. Prove that a conic circumscribing a triangle ABC will be an ellipse if the centre lie within the triangle DEF or within the angles vertically opposite to the angles of the triangle DEF, where D, E, F are the middle points of the sides of the triangle ABC.

18. Shew that the locus of the foci of parabolas to which the triangle of reference is self-polar is the nine-point circle. 19. Shew that the locus of the foci of all conics touching the four lines la±mß±ny = 0 is the cubic

P

+

P

+

P

+

P

0,

la + mẞ + ny la - mẞ - ny - la+mẞ-ny-la-mẞ+ny

where P22 + m2 + n2 − 2mn cos A - 2nl cos B-2lm cos C, and P2, P3, P have similar values.

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4

20. If a conic be inscribed in a given triangle, and its major axis pass through the fixed point (f, g, h), the locus of its focus is the cubic

21.

fa (ß3 − y3) + gẞ (y2 − a2) + hy (a3 − ß3) = 0.

If the centre of a conic inscribed in a triangle move along a fixed straight line, the foci will lie on a cubic circumscribing the triangle.

22. The locus of the centres of the rectangular hyperbolas with respect to which the triangle of reference is self-conjugate is the circumscribing circle.

23. The locus of the centres of all rectangular hyperbolas inscribed in the triangle of reference is the self-conjugate circle.

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, B, y respectively, then Aa, BB, 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 C'B' 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, P3, 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 p, p, to p ̧ ̧ will be constant.

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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.