Book V. c, and c to d, which are the fame, each to each, with the ratios of and Q to R. therefore, by the alfo let the ratio of A to B, that G to H, K to L, M to N, O to P, Becaufe e is to f, as (G to H, that is as) Y to Z; and f is to g, as (K to L, that is as) Z to a; therefore, ex aequali, e is to g, as Y to a. and, by the Hypothefis, A is to B, that is S to T, as e to g; wherefore S is to T, as Y to a, and, by inverfion, T is to S, as a to Y; and S is to X, as Y to d; therefore, ex aequali, T is to X, as a to d. alfo because h is to k, as (C to D, that is as) T to V; and k is to 1, as (E to F, that is as) V to X; therefore, ex aequali, h is to 1, as T to X. in like manner it may be demonstrated that m is to p, 11. 5. as a to d. and it was fhewn that T is to X, as a to d. therefore h is to 1, as m to p. Q. E. D. The Propofitions G and K are ufually, for the fake of brevity, expreffed in the fame terms with Propofitions F and H. and therefore it was proper to fhew the true meaning of them when they are fo expreffed; especially fince they are very frequently made use of by Geometers. THE Reciprocal figures, viz. triangles and parallelograms, are fuch as See N. "have their fides about two of their angles proportionals in such manner, that a fide of the first figure is to a fide of the other "as the remaining fide of this other is to the remaining fide of "the first." III. A straight line is faid to be cut in extreme and mean ratio, when the whole is to the greater fegment, as the greater fegment is to the lefs. IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to the bafe. K 2 PROP. Book VI. See N. TRIA PROP. I. THEOR. RIANGLES and parallelograms of the fame altitude are one to another as their bafes. Let the triangles ABC, ACD, and the parallelograms EC, CF have the fame altitude, viz. the perpendicular drawn from the point A to BD. then as the bafe BC is to the bafe CD, fo is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF. Produce BD both ways to the points H, L, and take any number of straight lines BG, GH, each equal to the base BC; and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, AL. then because CB, BG, GH are all equal, the tria. 38. 1. angles AHG, AGB, ABC are all equal. therefore whatever mul E A F HG B C D tiple the bafe HC is of the base BC, the fame multiple is the triangle AHC of the triangle ABC. for the fame reafon whatever multiple the bafe LC is of the bafe CD, the fame. multiple is the triangle ALC of the triangle ADC. and if the bafe HC be equal to the bafe CL, the triangle AHC is alfo equal to the triangle ALC; and K L if the bafe HC be greater than the bafe CL, likewife the triangle AHC is greater than the triangle ALC; and if lefs, lefs. therefore fince there are four magnitudes, viz. the two bafes BC, CD, and the two triangles ABC, ACD; and of the bafe BC and the triangle ABC the first and third, any cquimultiples whatever have been taken, viz. the bafe HC and triangle AHC; and of the bafe CD and triangle ACD the fecond and fourth have been taken any equimultiples whatever, viz. the base CL and triangle ALC; and that it has been fhewn that if the bafe HC be greater than the bafe CL, the triangle AHC is greater than b. 5. Def. 5. the triangle ALC; and if equal, equal; and if lefs, lefs. Therefore b as the bafe BC is to the base CD, fo is the triangle ABC to the triangle ACD. 41. 1. And because the parallelogram CE is double of the triangle ABC, and and the parallelogram CF double of the triangle ACD, and that Book VI. magnitudes have the fame ratio which their equimultiples have 4; as the triangle ABC is to the triangle ACD, fo is the parallelogram d. 15. 6. EC to the parallelogram CF. and because it has been fhewn that as the base BC is to the bafe CD, fo is the triangle ABC to the triangle ACD; and as the triangle ABC to the triangle ACD, fo is the parallelogram EC to the parallelogram CF; therefore as the bafe BC is to the base CD, fo is the parallelogram EC to the pa- c. 11. 5. rallelogram CF. Wherefore triangles, &c. Q. E. D. e COR. From this it is plain that triangles and parallelograms that have equal altitudes, are one to another as their bafes, Let the figures be placed fo as to have their bafes in the fame ftraight line; and having drawn perpendiculars from the vertices of the triangles to the bafes, the ftraight line which joins the vertices is parallel to that in which their bafes are f, because the perpendi- f. 33. 1. culars are both equal and parallel to one another. then, if the fame conftruction be made as in the Propofition, the Demonftration will be the fame. IF PROP. II. THEOR. Fa ftraight line be drawn parallel to one of the sides of see N. a triangle, it shall cut the other fides, or thefe produced, proportionally. and if the fides, or the fides produced be cut proportionally, the ftraight line which joins the points of fection fhall be parallel to the remaining fide of the triangle. Let DE be drawn parallel to BC one of the fides of the triangle ABC. BD is to DA, as CE to EA. Join BE, CD; then the triangle BDE is equal to the triangle CDE, because they are on the fame bafe DE, and between the a. 37. 1. fame parallels DE, BC. ADE is another triangle, and equal magnitudes have to the fame, the fame ratio; therefore as the triangle b. 7. 5. BDE to the triangle ADE, fo is the triangle CDE to the triangle ADE; but as the triangle BDE to the triangle ADE, fo is BD to c. 1. 6. DA, becaufe having the fame altitude, viz. the perpendicular drawn from the point E to AB, they are to one another as their bafes. and K 3 C for Book VI. for the fame reafon, as the triangle CDE to the triangle ADE, fo is CE to EA. Therefore as BD to DA; fo is CE to EA. Next, Let the fides AB, AC of the triangle ABC, or these pro d. 11. 5. duced, be cut proportionally in the points D, E, that is, fo that BD be to DA, as CE to EA; and join DE. DE is parallel to BC. The fame conftruction being made, because as BD to DA, so is CE to EA; and as BD to DA, fo is the triangle BDE to the tri. 1. 6. angle ADE; and as CE to EA, fo is the triangle CDE to the tri angle ADE; therefore the triangle BDE is to the triangle ADE, as the triangle CDE to the triangle ADE, that is, the triangles BDE, . 9. 5. CDE have the fame ratio to the triangle ADE; and therefore the triangle BDE is equal to the triangle CDE. and they are on the fame bafe DE; but equal triangles on the fame base are between f. 39. 1. the fame parallels f; therefore DE is parallel to BC. Wherefore if a ftraight line, &c. Q. E. D. IF PROP. III. THEOR. F the angle of a triangle be divided into two equal angles, by a ftraight line which alfo cuts the bafe; the fegments of the base shall have the fame ratio which the other fides of the triangle have to one another. and if the fegments of the base have the fame ratio which the other fides of the triangle have to one another, the straight line drawn from the vertex to the point of section divides the vertical angle into two equal angles. Let the angle BAC of any triangle ABC be divided into two equal angles by the ftraight line AD. BD is to DC, as BA to AC. Thro' |