Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/193

 the imilitude of the triangles EAF, ELI, ECH, EBG) AF is to LI as CH to BG. Likewie from the nature of the conic ections, LI (or CK) is to CD as CD to CH; and therefore ex æquo perturbate) AF is to CD, as CD to BG. Q. E. D.

Hence if two tangents FG, PQ meet two parallel tangents AF, BG in F and G, P and Q and cut one the other in O; AF (ex æquo pertubate) will be to BQ as AP to BG, and by diviion, as FP to GQ and therefore as FP to OG.

Whence alo the two right lines PG, FQ drawn through the points P and G, F and Q, will meet in the right line ACB, paing through the centre of the figure and the points of contact A, B.

If four ides of a parallelogram indefinitely produced touch any conic ection, and are cut by a fifth tangent; I ay, that taking those egments of any two conterminous ide which is intercepted between the point of contact and the third ide, is to the other egment. Pl. 11. Fig. 4.

Le the four ides ML, IK, KL, MI of the parallelogram MLIK touch the conic ection in