PROBLEMS
1. Two lines are drawn through a point on a conic so as always to make right angles with each other. Show that the lines joining the points where they meet the conic again all pass through a fixed point.
2. Two lines are drawn through a fixed point on a conic so as always to make equal angles with the tangent at that point. Show that the lines joining the two points where the lines meet the conic again all pass through a fixed point.
3. Four lines divide the plane into a certain number of regions. Determine for each region whether two conics or none may be drawn to pass through points of it and also to be tangent to the four lines.
4. If a variable quadrangle move in such a way as always to remain inscribed in a fixed conic, while three of its sides turn each around one of three fixed collinear points, then the fourth will also turn around a fourth fixed point collinear with the other three.
5. State and prove the dual of problem 4.
6. Extend problem 4 as follows: If a variable polygon of an even number of sides move in such a way as always to remain inscribed in a fixed conic, while all its sides but one pass through as many fixed collinear points, then the last side will also pass through a fixed point collinear with the others.
7. If a triangle QRS be inscribed in a conic, and if a transversal s meet two of its sides in A and A', the third side and the tangent at the opposite vertex in B and B', and the conic itself in C and C', then AA', BB', CC' are three pairs of points in an involution.
8. Use the last exercise to solve the problem: Given five points, Q, R, S, C, C', on a conic, to draw the tangent at any one of them.
9. State and prove the dual of problem 7 and use it to prove the dual of problem 8.
10. If a transversal cut two tangents to a conic in B and B', their chord of contact in A, and the conic itself in P and P', then the point A is a double point of the involution determined by BB' and PP'.
11. State and prove the dual of problem 10.
12. If a variable conic pass through two given points, P and P', and if it be tangent to two given lines, the chord of contact of these two tangents will always pass through a fixed point on PP'.
13. Use the last theorem to solve the problem: Given four points, P, P', Q, S, on a conic, and the tangent at one of them, Q, to draw the tangent at any one of the other points, S.
14. Apply the theorem of problem 9 to the case of a hyperbola where the two tangents are the asymptotes. Show in this way that if a hyperbola and its asymptotes be cut by a transversal, the segments intercepted by the curve and by the asymptotes respectively have the same middle point.
15. In a triangle circumscribed about a conic, any side is divided harmonically by its point of contact and the point where it meets the chord joining the points of contact of the other two sides.