# Apollonius' Construction of Tangents to Ellipses

Apollonius' recipe for construction tangent lines to ellipses and hyperbolas is as follows: [b]Proposition I-34[/b] Let $P$ be a point on an ellipse or hyperbola, $PB$ the perpendicular from the point to the main axis. Let $G$ and $H$ be the intersections of the axis with the curve and choose $A$ on the axis so that $\frac{AH}{AG}=\frac{BH}{BG}$. Then $AP$ will be tangent to the curve at $P$.

In the applet above, drag $H$ and/or $C$ to change the shape of the hyperbola. Drag $P$ to see the tangent line at any point on the hyperbola.