proportion among themſelves. Let the angles FGH, GHI, be ſo far increaſed that the right lines FG, GH, HI, may lie in direction, and by conſtructing the problem in this caſe, a right line fghi will be drawn, whoſe parts fg, gh, hi, intercepted between the four right lines given by poſition, AB and AD, AD and BD, BD and CE, will be one to another as the lines FG, GH, HI, and will obſerve the ſame order among themſelves. But the ſame thing may be more readily done in this manner.
Produce AB to K (Pl. 13. Fig. 2.) and BD to L, ſo as BK may be to B, as HI to GH; and DL to BD as GI to FG; and join KL meeting the right line CE in i. Produce iL to M, ſo as LM may be to iL as GH to HI; then draw MQ parallel to LB and meeting the right line AD in g, and join gi cutting AB, BD in f, h. I ſay the thing is done.
For let Mg cut the right line AB in Q and AD the right line KL in S, and draw, AP parallel to BD, and meeting iL in P, and gM to Lb (gi to bi, Mi to Li, GI to HI, AK to BK) and, AP to BL will be in the ſame ratio. Cut DL in R, ſo as DL to RL may be in that ſame ratio; and becauſe gS to gM, AS to AP, and DS to DL are proportional; therefore (ex æquo) as gS to Lb, ſo will AS be to BL, and DS to RL; and mixtly BL - RL to Lh - BL, as AS - DS to gS - AS. That is, BR is to Bh, as AD is to Ag, and therefore as BD to gQ. And alternately BR is to BD, as Bh to gQ, or as fh to fg. But by conſtruction the line BL was cut in D and R, in the ſame ratio as the line FI in G and H; and therefore
