is to PT, as DE to DH. Likewiſe PR is to DF as RG to DC, and therefore as (IG or) PS to DG; and, by permutation, PR is to PS as DF to DG; and, by compounding thoſe ratio's, the rectangle PQ x PR will be to tie rectangle PS x PT as the rectangle DE x DF is to the rectangle DG x DH, and conſequently in a given ratio. But PQ and PS are given, and therefore the ratio of PR to PT is given. Q. E. D.
Case 2. But if PR and PT are ſuppoſed to be in a given ratio one to the other, then by going back again by a like reaſoning, it will follow that the rectangle DE x DF is to the rectangle DG x DH in a given ratio; and ſo the point D (by lem. 18.) will lie in a conic ſection paſſing thro' the points A, B, C, P, as its locus. Q. E. D.
Cor 1. Hence if we draw BC cutting PQ in r, and in PT take Pt to Pr in the ſame ratio which PT has to PR: Then Pr will touch the conic ſection in the point B. For ſuppoſe the point D to coaleſce with the point B, ſo that the chord BD vaniſhing, BT ſhall become a tangent, and CD and BT will coincide with CB and Br. Co tt. z. And vice verla, if B r is atangent, and the lines B D, C D meet in any point D of a conic ſection; P R will be to P Tas Pr to Pr. And on the contrary, if.PR is to PTaſPr to Pr, (hm BD, and CD will meet in ſome point D of a conic ſection.
Cor. 2. One conic ſection cannot cut another conic ſection in more than four points. For, if it is poſſible, let two conic ſections paſs thro' the five points A, B, C, P, O; and let the right line BD cut them in the points D, d, and the right line Cd cut the right line PQ in q. Therefore PR is to PT as Pq to PT:
