The Conics of Apollonius

A. By a similar argument, the same F E will meet the diameter D C on the side of C.

Corollary. E G is greater than F H.

Prop. XXIV.

Fig. 34.

If a straight line (C E) meets a parabola or a hyperbola at one point (D) and, when extended in both directions, falls outside the section, it will meet the diameter (A B).

a 22. of this.

Let any point F be taken on the section, and let F D be joined; a this will meet the diameter; let us say at A; but the line C D crosses this (at D). Therefore, C D, when extended, will meet the diameter; namely, between A and the section.

Prop. XXV.

Fig. 35.

If a straight line (E F) meets an ellipse (at G) between two diameters (A B, C D) and, when extended in both directions, falls outside the section, it will meet both diameters.

Let G K be applied as an ordinate; this is parallel to the diameter A B: therefore E F will meet A B. By a similar argument, F E will meet C D.

Prop. XXVI.

Fig. 36.

If in a parabola or hyperbola a straight line (C D) is drawn parallel to the section's diameter (A B), it will meet the section at only one point (E).

That C D will meet the section is clear (since the distance of the parallel lines C D and A B is finite, but the section can be increased to infinity; and thus any ordinate drawn from A B to the section will exceed that distance). Let it meet at E; and let E F be applied as an ordinate: therefore, since all ordinates on the side of D a are larger than E F, and those on the side of C are smaller, it is clear that C D meets the section nowhere except at E.
a cor. 20. & cor. 21. of this.

Prop. XXVII.

Fig. 37.

If a straight line (C D) cuts the diameter of a parabola (A B), then when extended in both directions, it will meet the section.

Let A E be parallel to the ordinates; if C D is parallel to this, it is clear