other in the given ratio, and from AX cut off AC equal to one of them, and CD equal to the other; join D, B; and from C draw (E. 31. 1.) CE parallel to DB: Then is AB divided in E, so that the squares of AE and EB are to one another in the given ratio. For (constr. and E.2.6.) AE : EB :: AC :CD; .. (S. 1. 6. cor. 1.) AE*: EB:: AC2:CD2: But (constr.) AC is to CD in the given ratio; .. AE is to EB2 in the given ratio. PROP. XXIII. 31. PROBLEM. To find two points, situated in two adjacent sides of a given oblong, at equal distances from two opposite angles, from which, if two straight lines be drawn parallel to the sides of the figure, they shall cut off from it any part required. Let ABCD be a given oblong: It is required to find in two of its adjacent sides, as in AB and BC, two points equidistant from the A and C, from which if straight lines be drawn parallel to BC and BA, they shall cut off a given part of the oblong ABCD. From AB cut off AE-BC; produce CB; find (S. 21. 6.) a square which shall be the same part of the given oblong, as that which is to be cut off, and in CB, produced, make BF equal to the side of that square; bisect EB in G; from the centre G, at the distance GF, describe the circle HFL, cutting AB produced in L, and AB in H; from CB cut off CM-AH: Then are H and M the points which were to be found. For, since (constr.) AE= BC, and AH = CM, .., HE=BM: Again, sincé (constr.) HG=GL and EG=GB, .. HE=BL; and it has been shewn that HE=BM; .. BL=BM; but (constr. and E. 35. 3.) HBX BL=BF'; .. HBX BM = BF'; .. (constr.) HBXBM is the required part of ABX BC; and the points H and M are equidistant from A and C. PROP. XXIV. 32. PROBLEM. Within a given oblong, to describe another oblong which shall be any required part of it, and shall have its four sides all equally distant from the four sides of the given rectangle. Let ABCD be a given oblong: It is required to describe within it another oblong, which shall be a given part of ABCD, and have its sides equally distant from the sides of ABCD, each from each. From ABCD cut off (S. 23. 6.) the oblong HBMV, the same part of it as that which is to be described, is required to be, and having the extremities H and M, of its sides BH and BM, equally distant from A and C; let HV, produced, meet DC in I; bisect (E. 10. 1.) DI in K, and CM in N; ... DK = CN; from K draw (E. 31. 1.) KQ parallel to BC, and from N draw NR parallel to AB; from BC and NR cut off BP and NS, each equal to DK or CN; through P draw PQ parallel to AB, and through S draw ST parallel to BC; .. the figure QRST is an oblong: And it is manifest, from the construction, that RS=HB and RQ=BM, and that, .., the gnomon QRW is equal to the gnomon HBW, for the one may evidently be applied to the other so as to coincide with it; add to these equals the rectangle VT, and it is plain that the oblong QRST is equal to the oblong HM, which was made the required part of the given oblong ABCD. PROF. XXV. 33. PROBLEM. The base, the vertical angle, and the ratio of the two sides of a triangle being given, to construct it. Let EF be a given straight line: Upon EF, as K E H G a base, it is required to construct a A, having its vertical equal to a given, and its two remaining sides in a given ratio to one another. Upon EF describe (E. 33. 3.) a segment of a circle EKF, capable of containing an equal to the given, and complete the circle EKFG ; divide (E. 10. 6.) EF in H, so that EH is to HF in the given ratio; bisect (E. 30. 3.) ÉGF in G ; draw GH, and produce it to meet the circumference in K; lastly, join E, K, and F, K: Then is EKF the A which was to be constructed. For since (constr.) EG=FG, .. (E. 27. 3.) the < EKG = 2 FKG, so that the EKF is bisected by KH; .. (E. 3. 6.) KE:KF::EH:HF: That is (constr.) KE is to KF in the given ratio, and the vertical EKF is equal to the given angle. PROP. XXVI. 34. PROBLEM. A given finite straight line being divided into any two given parts, to divide it again, so that the rectangle contained by the two former given parts shall have a given ratio to the rectangle contained by the two latter parts. Describe (S. 21. 6.) a square which shall be to the rectangle, contained by the given parts of the given line, in the given ratio; and divide (S. 71.-3.) |