Book VI. PROP. XIV. THEOR. EQUAL parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional : and parallelograms that have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another. Let AB, BC be equal parallelograms, which have the angles at B equal, and let the sides DB, BE be placed in the same a 14. 1. straight line ; wherefore also FB, BG are in one straight line a : the sides of the parallelograms AB, BC about the equal angles, are reciprocally proportional ; that is, DB is to BE, as GB to BF. Complete the parallelogram FE ; and because the parallelogram AB is equal to BC, and that A F E c 1. 6. BE • ; and, as BC to FE, so is the D B d 11. 5. to BE, so is GB to BFd. Where fore the sides of the parallelograms G C are reciprocally proportional. But, let the sides about the equal angles be reciprocally proportional, viz. as DB to BE, so GB to BF; the parallelogram AB is equal to the parallelogram BC. Because, as DB to BE, so is GB to BF; and as DB to BE, so is the parallelogram AB to the parallelogram FE ; and as GB to BF, so is the parallelogram BC to the parallelogram FE ; there fore as AB to FE, so BC to FEd: wherefore the parallelogram e 9.5. AB is equal to the parallelogram BC. Therefore, equal paral lelograms, &c. Q.E.D. Book VI. PROP. XV. THEOR. EQUAL triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional : and trian. gles 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. 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 sides CA, AD be in one straight line; wherefore also EA and AB are in one straight line a ; and join BD. Because the triangle ABC is equal to the a 14. 1. triangle ADE, and that ABD is B D another triangle; therefore as the triangle CAB is to the trian. gle BAD, so is triangle EAD to triangle DABb: but as triangle 6 7. 5. CAB to triangle BAD, so is the A base CA to AD c; and as trian c 1.6 gle EAD to triangle DAB, so is the base EA to ABC; as there с E fore CA to AD, so is EA to AB d: d. 11.5. wherefore the sides 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 BADc; and as EA to AB, so is triangle EAD to triangle BAD«; therefore d as triangle BAC to triangle BAD, so is triangle EAD to triangle BAD; that is, the triangles BAC, EAD have the same ratio to the triangle BAD: wherefore the triangle ABC is equalee 9. to the triangle ADE. Therefore, equal triangles, &c. Q. E. D. Book VI. PROP. XVI. THEOR. IF four straight lines be proportionals, the rectangle 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. a 11. 1. From the points A, C draw a AG, CH at right angles to AB, CD; and make AG equal to F, and CH equal to E, and com plete the parallelograms BG, DH: because as AB to CD, so is b 7. 5. 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 but parallelograms which have their sides about equal angles 14.6. reciprocally proportional, are equal to one another c; therefore the parallelogram BG is equal to the parallelogram DH : and the parallelogram BG is contained E by the straight lines AB, F, be H F G в с D The same construction being made, because the rectangle contained by the straight 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 equiangu. lar: but the sides about the equal angles of equal parallelograms Book VI. are reciprocally proportionalc: wherefore, as AB to CD, so is CH to AG; and CH is equal to E, and AG to F: as therefore AB is c 14. 6. to CD, so E to F. Wherefore, if four, &c. Q. E. D. IF 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 square 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 isa to B, as D to C: but if four straight lines a 7.54 be proportionals, the rectangle contained by the A extremes is equal to that B which is contained by the D means b: therefore the C b 16. 6, rectangle contained by A, C is equal to that con C D tained by B, D. But the rectangle contained by B, D is the square of B; be A B cause 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 same construction being made, because the rectangle contained by A, C is equal to the square of B, and the square of B is equal to the rectangle contained by B, D, because B is equal to D; 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 proportionals b; therefore A is to B, as D to Book VI. C; but B is equal to D; wherefore as A to B, so B to C. There fore, if three straight lines, &c. Q. E. D. PROP. XVIII. PROB. See N. UPON a given straight line to describe a rectili. neal figure similar, and similarly situated to a given rectilineal figure. K Let AB be the given straight line, and CDEF the given rectilineal figure of four sides; it is required upon the given straight line AB to describe a rectilineal figure similar, and similarly situ. ated to CDEF. Join DF, and at the points A, B, in the straight line AB, a 23. 1. make a the angle BAG equal to the angle at C, and the angle ABG equal to the angle CDF; therefore the remaining angle b 32. 1. CFD is equal to the remaining angle AGB b: wherefore the triangle FCD is e- H F E L B C D about the equal angles proportionals: because the triangles GAB, c 4.6. 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, so, by reason of the equiangular triangles BGH, DFE, is FD to d 22. 5. FE; therefore, ex æqualid, 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 |