as bk x bn to bf x bd, ſo (by Lem. I7.) pq x pr to ps x pt; and ſo (hy ſuppoſition) PQ x PR to PS x PT. And becauſe of the ſimilar trapezia bkAf, PQAS, as bk to bf; ſo PQ to PS. Wherefore by dividing the terms of the preceding proportion by the correſpondent terms of this, we ſhall have bm to bd as PR to PT. And therefore the equiangular trapezia Dubd, DRPT are ſimilar, and conſequently their diagonal Dk, DP do coincide. Wherefore b falls in the interſection of the right lines AP, DP, and conſequently coincides with the point P. And therefore the point P where-ever it is taken falls to be in the aſſigned conic ſection. Q. E. D.
Cor. Hence if three right lines PQ, PR, PS, are drawn from a common point P to as many other right lines given in poſition AB, CD, AC, each to each, in as many angles reſpectively given, and the rectangle PQ x PR under any two of the lines drawn be to the ſquare of the third in a given ratio: The point P, from which the right lines are drawn, will be placed in a conic ſection that touches the lines AB, CD in A and C; and the contrary. For the poſition of the three right lines AB, CD, AC remain the ſame, let the line BD approach to and coincide with the line AC; then let the line PT come likewiſe to coincide with the line PS; and the rectangle PS; PT will become , and the right lines AB, CD, which before did cut the curve in the points A and B, C, and D, can no longer cut, but only touch, the curve in thoſe co-inciding points.
