# Apollonius' method for constructing tangent lines to conics

Apollonius' recipes for constructing tangent lines to parabolas, ellipses, and hyperbolas are as follows: [b]Proposition I-33[/b] Let$P$ be a point on the parabola with vertex $E$, with $PD$ perpendicular to the axis of symmetry. If $A$ is on the axis of symmetry so that $AE=ED$, then $AP$ will be tangent to the parabola at $P$. [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 $P$.}

Use Apollonius' method to construct the tangent lines of the given curves by first determining the x intercept (for ellipses and hyperbolas) or the y intercept (for parabolas) of the tangent line according to Apollonius' propositions.