Book VI. PROP. XV. THEOR. E QUAL triangles which have one angle of the one equal to one angle of the other, have their fides about the equal angles reciprocally proportional. and triangles which have one angle in the one equal to one angle in the other, and their sides about the equal angles reciprocally proportional, are equal to one another. b. 7. 5• E C. 1.6. Let ABC, ADE be equal triangles which have the angle BAC equal to the angle DAE; the sides about the equal angles of the triangles are reciprocally proportional; that is, CA is to AD, as EA to AB. Let the triangles be placed so that their fides CA, AD be in one straight line; wherefore also EA and AB are in one straight line a; and join BD. Because the 2. 14. 1. B D triangle ABC is equal to the triangle ADE, and that ABD is another triangle; therefore as the triangle CAB is to the triangle BAD fo is triangle А EAD to triangle DAB b. but as triangle CAB to triangle BAD, so is the base CA to ADC; and as triangle C C. 1. 6, EAD to triangle DAB, so is the base EA to AB “; as therefore CA to AD, so is EA to AB d. wherefore d. 11. So the fides of the triangles ABC, ADE about the equal angles are reciprocally proportional. But let the sides of the triangles ABC, ADE about the equal angles be reciprocally proportional, viz. CA to AD, as EA to AB; the triangle ABC is equal to the triangle ADE. Having joined BD as before, because as CA to AD, so is EA to AB; and as CA to AD, so is triangle BAC to triangle BAD , and as EA to AB, so triangle EAD to triangle BADs; therefore d as triangle BAC to triangle BAD, fo is triangle EAD to triangle BAD; that is, the triangles BAC, EAD have the faine ratio to the triangle BAD. wherefore the triangle ABC is equal e to the tri-c. 9. S. angle ADE. Therefore equal triangles, &e. Q. F. D. Book vi PROP. XVI. THEOR. 3. 11. I. 6. 7. 5 €. 14. 6. IF contained by the extremes is equal to the rectangle contained by the means, and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals. Let the four straight lines AB, CD, E, F be proportionals, viz. as AB to CD, so E to F; the rectangle contained by AB, F is equal to the rectangle contained by CD, E. From the points A, C draw : AG, CH at řight angles to AB, CD; and make AG equal to F, and CH equal to E, and complete the parallelograms BG, DH. because as AB to CD, so is E to F; and that E is equal to CH, and F to AG; AB is b to CD, as CH to AG. therefore the sides of the parallelograms BG, DH about the equal angles are reciprocally proportional; but parallelograms which have their fides about equal angles reciprocally proportional, are equal to one another°; therefore the patallelogram BG is equal to the parallelogram DH. and the parallelogram BG is con E H A B C D And if the rectangle contained by the straight lines AB, F be equal to that which is contained by CD, E; these four lines are proportionals, viz. AB is to CD, as E to F. The fame conftruction being made, because the rectangle contained by the ftraight lines AB, F is equal to that which is contained by CD, E, and that the rectangle BG is contained by AB, F, because AG is equal to F; and the rectangle DH by CD, E, because CH is equal to E; therefore the parallelogram BG is equal to the parallelogram DH; and they are equiangular. but the fides about the equal angles of equal parallelograms are reciprocally pro- Book VI. portional, wherefore as AB to CD, fo is CH to AG; and CH is equal to E, and AG to F. as therefore AB is to CD, fo E to F. C. 14 6. Wherefore if four, &¢. Q. E. D. PROP. XVII. THEOR. F three straight lines be proportionals, the rectangle contained by the extremes is equal to the square of the mean: and if the rectangle contained by the extremes be equal to the square of the mean, the three straight lines are proportionals. Let the three straight lines A, B, C be proportionals, viz. as A to B, so B to C; the rectangle contained by A, C is equal to the Squase of B. Take D equal to B; and because as A to B, so B to C, and that B is equal to D; A is a to B, as D to C. but if four straight lines a. 7.50 be proportionals, the rectangle contained by the Aextremes is equal to that B which is contained by the D means, therefore the rec b. 16. & tangle contained by A, C is equal to that contained C D by B, D. but the rectangle contained by B, D is A B the square of B; because B is equal to D. therefore the rectangle contained by A, C is equal to the square of B. And if the rectangle contained by A, C be equal to the square of B; A is to B, as B to C, The fame construction being made, becaufe the rectangle contained by A, C is equal to the fquare of B, and the square of B is equal to the rectangle contained by B, D, because B is equal to Di therefore the rectangle contained by A, C is equal to that contained by B, D. but if the rectangle contained by the extremes be equal to that contained by the means, the four straight lines are propora tionals b, therefore A is to B, as D to C; but B is equal to Di Book VI. wherefore as A to B, fo B to C. Therefore if three straight Comprend lines, &c. Q. E. D. UPON TPON a given Araight line to describe a re&ilineal figure similar, and similarly situated to'a given rectilineal figure. Let AB be the given straight line, and CDEF the given rectilinéal figure of four fides; it is required upon the given straight line AB to describe a rectilineal figure fimilar and similarly fituated to CDEF. A. 23. 1. Join DF, and at the points A, B in the straight line AB make a the angle BAG equal to the angle at C, and the angle ABG equal to the angle CDF; therefore the remaining angle CFD is equal to the 6. 32. I. remaining angle AGB 6. wherefore the triangle FCD is equiangular to the triangle GAB. F K к GBH equal to FDE; therefore the remaining A B C D angle FED is equal to the remaining angle GHB, and the triangle FDE equiangular to the triangle GBH. then because the angle AGB is equal to the angle CFD, and BGH to DFE, the whole angle AGH is equal to the whole CFE. for the same reason, the angle ABH is equal to the angle CDE; also the angle at A is equal to the angle at C, and the angle GHB to FED. therefore the rectilineal figure ABHG is equiangular to CDEF. but likewise these figures have their Gides about the equal angles proportionale. because the triangles GAB, FCD being equiangular, BA is c to AG, as DC to CF, and because AG is to GB, as CF to FD; and as GB to GH, fo, by reason of the equiangular triangles BGH, DFE, is FD to FE; d. 22. 5. therefore, ex aequali“, AG is to GH, as CF to FE. in the same manner it may be proved that AB is to BH, as CD to DE. and GH is to HB, as FE to ED ¢, Wherefore because the rectilineal C. 4. 6. figures ABHG, CDEF are equiangular, and have their fides about Book VI. the equal angles proportionals, they are similar to one another Next, Let it be required to describe upon a given straight line 4. 1. Def. G. AB, a rectilineal figure similar, and similarly situated to the recti, lineal figure CDKEF of five sides. Join DE, and upon the given straight line AB describe the rectilineal figure ABHG similar and similarly situated to the quadrilateral figure CDEF, by the former case. and at the points B, H in the straight line BH, make the angle HBL equal to the angle EDK, and the angle BHL equal to the angle DEK; therefore the remaining angle at K is equal to the remaining angle at L. and because the figures ABHG, CDEF are similar, the angle GHB is equal to the angle FED, and BHL is equal to DEK; wherefore the whole angle GHL is equal to the whole angle FEK. for the same reason, the angle ABL is equal to the angle CDK. therefore the five sided figures AGHLB, CFEKD are equiangular. and because the figures AGHB, CFED are similar, GH is to HB, as FE to ED; and as HB to HL, fo is ED to EK“; therefore ex c. 4. 6. aequalid, GH is to HL, as FE to EK. for the same reason, AB is d. 22. 5. to BL, as CD to DK. and BL is to LH, as · DK to KE, because the triangles BLH, DKE are equiangular. therefore because the five faded figures AGHLB, CFEKD are equiangular, and have their fides about the equal angles proportionals, they are similar to one another. and in the same manner a rectilineal figure of fix fides may be described upon a given straight line similar to one given, and so on. Which was to be done. IMILAR triangles are to one another in the du. plicate ratio of their homologous fides. Let ABC, DEF be similar triangles having the angle B equal to the angle E, ånd let AB be to BC, as DE to EF, so that the Gide BC is homologous to LF a. the triangle ABC has to the triangle a. 12. Def.sa DEF, the duplicate ratio of that which BC has to EF. Take BG a third proportional to BC, EF b, fo that BC is to EF, b. 11, 6. as EF to BG, and join GA. then, because as AB to BC, so DE to EF; alternately “, AB is to DE, as BC to EF. but as BC to EF, so 6. 16. Sa |