given; wherefore IEF, and the segments PG and OF are given. COMPOSITION. Make PKM and OL= N, join KL, PO, and produce them to meet in I, and draw IEF; PG and OF are the required segments. For (VI. 2. El.) the parallels AB and CD being cut proportionally by the diverging lines IK, IP, and IG,-PG is to OF as KP to OL, that is, as M to N. If M be equal to N, the point I vanishes, and EF becomes evidently a parallel to OP. If the straight lines KL and PO meet in the given point E, the problem is by its nature indeterminate, or it admits of indefinite solution; for, in that case, the segments PE and OF, intercepted by any straight line whatever, drawn through D, have all the same ratio. PROP. II. PROB. Two diverging lines being given in position, to draw, through a given point, a straight line intercepting segments which shall have a given ratio. Let it be required, through D, to draw EDF, so that AE shall be to AF in the ratio of M to N. since GD and the point G are evidently given, GF and the point Fare likewise given. COMPOSITION. From AB and AC cut off AK = M, and AL = N, join KL, and parallel to it draw EDF through D; AE and AF are the segments required. For (VI. 1. El.) the parallels EF and KL cut the diverging lines AB and AC proportionally, and therefore AE is to AF, as AK to AL, that is, as M to N. PROP. III. PROB. Two diverging lines being given in position, to draw, through a given point, a straight line cutting off segments on the one from their intersection, and on the other from a given point-that shall have a given ratio. Let AB and AC be two diverging lines, it is required, through the point D, to draw EDF, so that AE shall be to the part OF, in the ratio of M to N. ANALYSIS. Draw DG parallel to AE, and meeting AC, or its production in G, and make AE: GD :: OF: OH. By alternation, AE; OF:: GD: OH; but the ratio of AE to OF is given, and thence that of GD to OH; and since GD and the point G are given, OH and the point H are also given. Again, because AE: GD: OF: OH, and (VI. 2. El.) AE : GD :: AF: GF, it follows that OF:OH::AF:GF; whence (V. 10. El.) FH: OH :: AG GF, and (V. 6. El.) GF.FH AG.OH. But AG and OH are both given, and consequently the rectangle un : E B D A OG H F C der the segments OF and FH of the given portion GH is also given, and thence the point of section F is given, and the straight line EDF. COMPOSITION. Make GD to OH, as M to N, and (VI. 20. El.) divide GH in F, so that the rectangle GF, FH shall be equal to AG,OH, and draw EDF; then the segment AE is to OF as M to N. Since GF.FH AG.OH, therefore FH: OH :: AG: GF, and (V. 10. El.) OF: OH :: AF: GF: but (VI. 2. El.) AE : GD :: AF: GF, and consequently AE: GD :: OF : OH, and alternately AE: OF :: GD: OH, that is in the given ratio. PROP. IV. PROB. Two diverging lines being given in position, to draw, through a given point, a straight line, cutting off segments from given points in a given ratio. Let AB and AC be two diverging lines; it is required, through the point D, to draw EDF, so that PE shall be to OF in the ratio of M to N. is given, whence IK, being parallel to AB, is likewise given in position. But (VI. 2. El.) PE : IK :: PD : ID, and since PD and ID are both given, the ratio of PE to IK is given; consequently, the ratio of PE to OF being given, the ratio of IK to OF is given. Wherefore, by the last proposition, the straight line KDF is given in position. COMPOSITION. Join PD and draw IK parallel to AB, make M to L, as PD to ID, and draw, by the last proposition, KDF, so that IK shall be to OF, as L to N; then will PE and OF be the segments required. For (VI. 2. El.) PE: IK :: PD: ID :: M: L, and IK: OF :: L: N; whence (V. 16. El.) PE: OF :: M: N. PROP. V. PROB. Two parallels being given, from a point in a given intersecting line, to draw another straight line cutting off segments which shall contain a given rectangle. Let AB, CD be two parallels, and G a given point, through which it is required to draw FE intercepting, from given points O and P in the same direction OPG, segments OE and PF, that shall contain a given rectangle. angle OE, PF is given, and hence the square of OE, and OE itself, are given. COMPOSITION. Find (VI. 18. El.) GI, a mean proportional between GO and GP, draw IK parallel to AB or CD, and such (III. 37. |