Apollonius' method for constructing tangent lines to conics
Apollonius' recipes for constructing tangent lines to parabolas, ellipses, and hyperbolas are as follows:
Proposition I-33 Let be a point on the parabola with vertex , with perpendicular to the axis of symmetry. If is on the axis of symmetry so that , then $AP$ will be tangent to the parabola at .
Proposition I-34 Let be a point on an ellipse or hyperbola, the perpendicular from the point to the main axis.
Let and be the intersections of the axis with the curve and choose on the axis so that . Then will be tangent to the curve at .}