Trapezium

Lemma 27

A trapezium given in kind, the angles whereof may be in placed, respect of four right lines given by position, that are neither all paralhl among themselves, nor converge to one common point, describe , the several angles may touch the several lines. Let the four right lines ABC, AD, BD, CE, be given by position the first cutting the second in A, the third in B, and the fourth in C and suppose a ; trapezium fghi is to be described that may be similar trapezium FCHI, and whose angle /, equal to the given angle F, may touch the right line ABC and to the ; other angles g, h, i, equal to the other given angles, G, H, I, may touch the other lines AD, BD, CE, re Join FH, and upon FG. FH, FI describe spectively. (lie as of J% many segments of circles FSG, FTH, FVI, the first which FSG may be capable of an angle equal to BAD FTH the second the angle capable of an angle the third and FVI of an angle equal to the angle to the equal angle the segments are to be described towards those sides of the ACE. ; CBD ; Bnrf>, lines the FG, FH, same FI, that the circular order of the letters BADB, and that the letters .ibout in the same order FSGF may FTHF as of the letters as the letters CBDC and the letters may be turn FVIF in the game order as the letters ACE A. Complete the segments into entire cir cles, and let P be the centre of the first circle FSG, Q, the centre of the Join and produce both ways the line PQ,, and in it take second FTH. QR in the same ratio to PQ as BC has to AB. But QR is to be taken towards that side of the point Q that the order of the letters P, Q,, ROF NATURAL PHILOSOPHY. SEC. V.J may be the same as of the letters A, B, C with the interval ; R and about the centre RF describe a fourth circle third circle (lie FVI in FNc Join c. cutting Fc cut 1 and the second in and let the figure first circle in a, ting the Draw aG, &H, be made / . 15] ABC/ 4f/ii cl, similar to the figure w^cFGHI; and the trapezium fghi will be that which was required to be de scribed. For let the two RK, The "K, FSG, FTH join PK, Q,K, first circles K cut one the other in ; QP 6K, cK, and produce angles FaK, F6K, FcK to L. at the circumferences are the halves of the FPK, FQK, FRK, LRK, the halves of at the centres, and therefore equal to LPK, those angles. Wherefore the figure is to the figure and and similar ab is be to 6cK, consequently iquiangular to BC. But by construction, the angles res PQ, to Q,R, that is, as angles LQ.K, PQRK AB /B//,/C? are equal Air, , to the angles FG, F&H, Fcl. And therefore be completed similar to the figure abcFGHl. vVliich done a trapezium fghi will be constructed similar to the trapezium FGHI, and which by its angles/, g, h, i will touch the right lines ABC, the figure ABCfghi may AD, BD, CE. Q.E.F. COR. Hence a right line may be drawn whose parts intercepted in a given order, between four right lines given by position, shall have a given Let the angles FGH, GHI, be so far in proportion among themselves. creased that the right lines FG, GH, HI, may lie in directum ; 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 AD, AD and BD, BD and CE, will be one to another lines FG, GH, HI, and will observe the same order among them But the same thing may be more readily done in this manner. AB position, as the selves. Produce so as GH BK LM AB may to K BD to L, AB as HI to BD as GI to FG; and be to DL to KL meeting and ; and join CE and the right line Produce iL to iL as be may in i. to M, GH II so as to HI ; draw MQ, and AD and M* join gi cutting AB, BD in f, h I then parallel to LB, in g, meeting the right line ; say, the thing is done. For let MO- cut the right line AB in Q, and AD the right line KL iuTHE MATHEMATICAL PRINCIPLES ^52 to hi, AP draw S, arid Mi to Li, to in 11, DL Cut ratio. parallel to GI AS cause ffS to g~M, (ex ceqit.o) gS RL to as AD is to BD as BL Eh to as Lh D BR therefore it is and //.. BK) to AP. and RL DS and AP may to in P, and -M to Lh (g
BL, will be in the same to be in that same ratio; and be DL are proportional; therefore AS be to BL, and DS to RL and mixtly. AS DS to gS AS. That is, BR is to LA, so will BL, as ; Ag, and therefore as BD to gQ. And alternately BR is But by construction the line BL or as fh to fg. and R in the same ratio as the line FI in G and H and Wherefore fh is to fg as FH to is to BD as FH to FG. ; Since, therefore, HI, DL to I. 13/i to g-Q,, was cut in FG. to HI, so as to BD, and meeting iL AK [BOOK gi to hi likewise is as Mi to Li, that manifest that the lines FI, fi, are similarly cut in GI is, as G and H, to g Q.E.F. LK is drawn cutting Ei as FH to HI, In the construction of this Corollary, after the line in i, we may produce iE to V, so as may be to EV CE then draw V/~ parallel to BD. It will come to the same, if about the in X, and i with an interval IH, we describe a circle cutting so as iY may be equal to IF, and then draw Yf parallel produce iX to arid BD centre Y to BO. Sir Christopher Wren and Dr. Wallis have long ago given other solu tions of this Problem.