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
Fig. 39