The Conics of Apollonius

Prop. VII.

Fig. 11.

If a cone is cut by a plane (ABC) through the axis, and is also cut by another plane (DFE) cutting the plane of the base of the cone along a straight line (DE) which is perpendicular either to (BC), the base of the triangle through the axis, or to that which is constituted in direct alignment with it: the lines (HK) which are drawn from the section (DFE) made by the plane on the surface of the cone equidistant to that (DE), which is perpendicular to the base of the triangle (BC), will fall into the common section (FG) of the cutting plane and the triangle through the axis. And if the cone is a right cone, the line (DE), which is in the base, will be perpendicular to (FG), the common section of the cutting plane and the triangle through the axis; but if it is Scalene, not always, unless when the plane (ABC), which is led through the axis, is at right angles to the base of the cone (BDCE).

a 4. def. 11. b 4. 11. d 18. 11.

That HK meets the plane ABC, and therefore its common section FG with the plane DFE, and is bisected at that intersection, is clear from the preceding. Furthermore, if the cone is right, the circle BC will be at right angles to the plane ABC; and for that reason DE is at right angles to the plane ABC; and therefore DE is perpendicular to FG. The same argument holds if the triangle ABC is at right angles to the circle BC in any way; but if this is not the case, DE will not be perpendicular to FG: For if DE were perpendicular to both BC and FG, the same DE would be at right angles to the plane ABC; whence the circle BC would be at right angles to the triangle ABC, contrary to the Hypothesis.

Coroll. Hence FG is the diameter of the section DFE, inasmuch as it bisects the lines parallel to DE.

Prop. VIII.

Fig. 12.

If a cone is cut by a plane (ABC) through the axis, and is cut by another plane (DFE) cutting the base of the cone (BDC) along a straight line (DE), which is perpendicular to (BC), the base of the triangle through the axis; and the diameter (F G) of the section made on the surface either is equidistant to one of the sides (AC) of the triangle, or meets it outside the vertex of the cone; and both the surface of the cone (ABC) and the cutting plane (DFE) are produced to infinity; the section itself (DFE) will also be increased to infinity; and it will cut off from the diameter (FG) of the section to the vertex a line (MNH) equal to any given line (CFH), which line (MN) has been drawn from the conic section (MFN) equidistant to that (DE) which is in the base.