Lemma XXIIII.
If two right lines as AC, BD given by poſition, and terminating in given points A, B, are in a given ratio one to the other, and the right line CD, by which the indetermined points CD are joined, is cut in K in a given ratio; I ſay that the point K will be placed in a right line given by poſition. Pl. 11. Fig. 2.
For let the right lines AC, BD meet in E, and in BE take BG to AE, as BD is to AC, and let FD be always equal to the given line EG; and by conſtruction, EC will be to GD, that is, to EF, as AC to BD, and therefore in a given ratio; and therefore the triangle EFC will be given in kind. Let CF be cut in L ſo as CL may be to CF in the ratio of CK to CD; and becauſe that is a given ratio, the triangle EFL will be given in kind, and therefore the point L will be placed in the right line EL given by poſition. Join LK and the triangles CLK, CFD will be ſimilar; and becauſe FD is a given line, and LK is to FD in a given ratio, LK will be alſo given. To this let EH be taken equal, and ELKH will be always a parallelogram. And therefore the point K is always placed in the ſide HK (given by poſition) of that parallelogram. Q. E. D.
