Proposition XXVII. Problem XIX.
To deſcribe a trajectory that may touch five right lines given by poſition. Pl. 11 . Fig. 5.
ſuppoſing ABG, BCF, GCD, FDE, EA to be the tangents given by poſition. Biſect in M and N, AF, BE the diagonals of the quadrilateral figure ABFE contained under any four of them; and (by cor. 3. lem. 25) the right line MN drawn through the points of biſection will paſs through the centre of the trajectory. Again. biſect in P and Q the diagonals (if I may ſo call them) BD, GF of the quadrilateral figure BGDF contained under any other four tangents, and the right line PQ drawn through the points of biſection will paſs through the centre of the trajectory. And therefore the centre will be given in the concourſe of the biſecting lines. Suppoſe it to be O. Parallel to any tangent BC draw KL, at ſuch diſtance that the centre O may be placed in the middle between the parallels; this KL will touch the trajectory to be deſcribed. Let this cut any other two tangents GCD, FDE, in L and K. Through the points C and K, F and L, where the tangents not parallel CL, FK meet the parallel tangents CF, KL, draw CK, FL meeting in R; and the right line OR drawn and produced, will cut the parallel tangents CF, KL, in the points of contact. This appears from cor. 3. lem. 24. And by the ſame method the other points of contact may be found, and then the trajectory may be deſcribed by prob. 14. Q. E. F.
