Cor. Becauſe the figure EFLC is given in kind the three right lines EF, EL and EC, that is GD, HK and EC will have given ratio's to each other.
Lemma XXIV.
If three right lines, two whereof are parallel, and given by poſition, touch any conic ſection; I ſay, that the ſemidiameter of the ſection which is parallel to thoſe two is a mean proportional between the ſegments of thoſe two, that are intercepted between the points of contact and the third tangent. Pl. 11. Fig. 3.
Let AF, GB be the two parallels touching the conic ſection ADB in A and B; EF the third right line touching the conic ſection in I, and meeting the two former tangents in F and G, and let CD he the ſemi-diameter of the figure parallel to thoſe tangents; I ſay, that AE, CD, BG are continually proportional.
For if the conjugate diameters AB, DM meet the tangent FG in E and H, and cut one the other in C, and the parallelogram IKCL be compleated; from the nature of the conic ſections, EC will be to CA as CA to CL, and ſo by diviſion, EC - CA to CA - CL or EA to AL; and by compoſition, EA to EA + AL or EL, as EC to EC + CA or EB; and therefore (becauſe of
