You are here  Can You Really Derive Conic Formulae from a Cone?
- Deriving the Symptom of the Hyperbola  Gary S. Stoudt
Apollonius, Conics, Book I, Proposition 12   If a cone is cut by
a plane through its axis, and also cut by another plane cutting
the base of the cone in a straight line perpendicular to the
base of the axial triangle, and if the diameter of the section
produced meets one side of the axial triangle beyond the vertex
of the cone, then any straight line which is drawn from the
section to its diameter parallel to the common section of the
cutting plane and of the cone's base, will equal in square some
area applied to a straight line to which the straight line added
along the diameter of the section and subtending the exterior
angle of the triangle has a ratio that the square on the
straight line drawn from the cone's vertex to the triangle's
base parallel to the section's diameter has to the rectangle
contained by the sections of the base which this straight line
makes when drawn, this area ha