Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/194

 130 A, B, C, D; and let the fifth tangent FQ cut thoe ides in F, Q, H and E, and taking the egments ME, KQ of the ides MI, KI; or the egments KH, MF of the ides KL, ML; I ay, that ME is to MI as BK to KQ; and KH to KL, as AM to MF. For, by cor. 1. of the preceding lemma, ME is to EL as (AM or) BK to BQ; and, by compoition, ME is to MI as BK to KQ. Q. E. D. Alo KH is to HL as (BK or) AM to AF and by diviion KH to KL, as AM to MF. Q. E. D.

Hence if a parallelogram IKLM decribed about a given conic ection is given, the rectangle KQ×ME, as alo the rectangle KH×MF equal thereto, will be given. For, by reaon of the imilar triangles KQH, MFE, thoe rectangles are equal.

And if a ixth tangent eq is drawn meeting the tangents KI, MI in q and e; the rectangle KQ×ME will be equal to the rectangle Kq×Me, and KQ will be to Me, as Kq to ME, and by diviion as Qq to Ee.

Hence alo if Eq, eQ are joined and biected, and a right line is drawn through the points of biection, this right line will pas through the centre of the conic ection. For ince Qq is to Ee, as KQ to Me; the ame right line will pas through the middle of all the lines Eq, eQ, MK (by lem. 22.) and the middle point of the right line MK is the centre of the ection.