PROPOSITION V. THEOREM V. IF the fides of two triangles (A BC, DE F) be proportionals, thofe triangles fhall be equiangular, and have their equal angles (A & m, C & n, &c) opposite to the homologous fides (B C, E F & AB, DE, &c). Hypothefis. The A ABC, DEF have their fides proportionals, that is, SABAC : IAB BC AC: BC : DE DF. DF: EF. DE: EF. Thefis. 1. The AAB C, D E F are equiangular. II. The oppofite to the homologous fides are; or VAVm, V C = V n & VBV E. II. The fides BC, E F, A B, DE, A C, D F. are bomologous. Preparation. 1. At D in D F make VVA & at F, W9=VC. BECAUSE DEMONSTRATION. ECAUSE in the equiangular ▲ A B C, DGF (Prep. 1. & P. 32. 1. AB: ACDG: DF, & AB AC B. 6. 4. SP.11. B.5. P. 9. B. 5. 3.5. P. 8. B. 1. But the ADGF, is equiangular to the AABC (Prep.1. & P.32.B.1), 6. From whence it follows that the AABC, DEF are equiangular. Ax.1. B. 1. Which was to be demonftrated. 1. 7. Moreover, the VA, C & B oppofite to the fides B C, A B, A C, be- 8. It follows, that the VA, m; alfo C, n & B, E oppofite to the homo- Cor. Therefore thofe triangles are also fimilar. (D. 1. B. 6.) IF A C G PROPOSITION VI. THEOREM VI two triangles (A B C, D E F) have one angle (A) of the one equal to one angle (m) of the other, and the fides (BA, A C, & ED, D F), about the equal angles proportionals, the triangles fhall be equiangular, and shall have these angles (C & n, alfo B & E) equal which are oppofite to the homologous fides (BA, ED & AC, D F). Hypothefis. I. VA II. BA to m. ACED: DF. III. BA, ED; AC, DF are homologous. Thefis. 1. The ABC, DEF are equiangular. II. The C&n, also the VB & E are Preparation. to one anotber. 1. At the point D in the ftraight line D F make V to to VC. P. 23. B.1. Lem. B.1. 2. Produce the fides DG, F G until they meet in G. ECAUSE the AABC, DG F are equiangular (Prep.1. & P. 32. B. 1), & particularly VCVg & VB = Therefore the two A DEF, DG F having the two fides E D, D F to the two fides GD, DF (Arg. 3) &m top (Prep. 1). 4. The Vn, q & E, G are, & the A DEF, DGF are equiangular. P. 4. B.1. But the AABC, DGF being alfo equiangular (Prep.1. & P.32. B.1), 5. It follows, that the AA BC, DEF are equiangular. Which was to be demonftrated. 1. Moreover, each of the angles C & n being to V9 (Prep.1. 6. The VC is to V n. Ax. 1. B.1. Arg.4). Ax. 1. B.1. 7. Confequently, VA being to Vm (Hyp.1), V B is alfoto V E. P. 32. B.1. And the fides B A, ED & AC, D F oppofite to those angles being homologous (Hyp. 3. & D. 12. B. 5.). 8. It follows, that the VC & n, alfo B & E oppofite to those homologous fides are to one another. Which was to be demonftrated. 11. Cor. Therefore thofe triangles are alfo fimilar to each other. (P.4. Cor. B.6). IF PROPOSITION VII. THEOREM VII. F two triangles (A B C, D E F) have one angle of the one (B), equal to one angle of the other (E), and the fides (B A, AC & E D, DF) about two other angles (A & D), proportionals; then if each of the remaining angles (C & F) be either acute, or obtufe, the triangles fhall be equiangular, and have those angles (A & D) equal, about which the fides are proportionals. Hypothefis. 1. V B is to V E. II. BA: AC=ED:DF III. The VC & F are both either acute, or obtufe. Thefis. The ABC, DEF are equiangular, & the BAC & D are = to one another. DEMONSTRATION. If not, the VBAC & D are unequal, and one as BAC is . Wherefore, VC is to V m. 3. Confequently, BA: AGBA: AC, From whence it follows that AG is to A C. And because in this cafe YC is < L. 6. The Vm will be alfo <L; & Vn which is adjacent to it > L. But this Vn being to VF (Arg.1), which in this cafe is <.13. B.1. 7. This fame Vn will be alfo <L; which is impoffible. 8. The VBAC & D are therefore to one another, & the third C P.11. B.5. P. 9. B5. P. 5. B.I. is to F, or the ▲ A B C, D E F are equiangular. P.32. B.1. Which was to be demonftrated. By CASE II. If the VC & F are both obtufe. Y the fame reasoning as in the first Case (Arg.1. to Arg. 5.) it may be proved, that 1. The VC is to V m. 2. Therefore Vm is alfo > L, & the which is impoffible. 3. Confequently, the VBAC & D are VC is to VF, or the A A B C, D E F are equiangular. C+ will be > 2 L, P.17. B.I. to one another & the third P.32. B.1. Which was to be demonstrated. REMARK. If the F the C&F are both right angles the ▲ A B C & DEF are equiangu lar (Hyp. 1. & P. 32. B. 2). Cor. Therefore thofe triangles are fimilar to one another (P.4. Cor. B. 6). PROPOSITION VIII. THEOREM VIII. IN a right angled triangle (A B C), if a perpendicular (B D) be drawn from the right angle (A BC) to the bafe A C, the triangles (A D B, B D C) on each fide of it are fimilar to the whole triangle (A BC) and to one another. Hypothefis. 1. The AA B C is rgle. in B. II. B D is upon A C. BECAUSE Thefis. The AADB, BDC are fimilar to one another, & each is also fimilar to the whole ▲ A B C. DEMONSTRATION. ECAUSE in the two rgle. A A D B, A B C, the Vm is VABC, (Ax. 10. B. 1.), & VA common to the two A. = to 1. The Vo is to VC & the two ▲ A B C, A D B are equiangular. 2. Confequently, thofe two A are also similar. It may be demonstrated after the fame manner, that 3. The ABDC is fimilar to the ▲ A B C. P.32. B ́1. SP. 4. B. 6. {Cor. Likewife in the two rgle. A A D B, B DC, V being to n (Ax. 10. B. 1.) & Vo to VC (Arg. 1). SP. 4. B. 6. Cor. 4. The VA is to V p, & the two AA DB, BD C, are equiangular. P.32.B.1. 5. From whence it follows that these ▲ are fimilar. 6. Confequently, the 1BD divides the ▲ A B C into two ▲ ADB; BDC fimilar to one another (Arg. 5.) & fimilar to the whole ▲ ABC (Arg. 2. & 3). FROM Which was to be demonftrated. ROM this it is manifeft that the perpendicular B D drawn from the Vertex of a right angled triangle to the bafe, is a mean proportional between the fegments A D & D C of the bafe; for the triangles A DB, B DC being equiangular, AD: DB DB: DC (P. 4. B. 6.). Alfo, each of the fides A B or B C of the triangle A B C is a mean proportional between the bafe & and the fegment A D or DC adjacent to that fide. for fince each of the triangles A D B, BDC is equiangular with the whole AA BC, AC: AB ABAD, & AC: BC BC: DC (P. 4. B. 6). |