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. CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS