Apollonius' Construction of Tangents to Ellipses
Apollonius' recipe for construction tangent lines to ellipses and hyperbolas is as follows:
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 .