Take two straight lines DE, DF containing any angle EDF; and Book VI. upon these make DG equal to A, GE equal to B, and DH equal to C; and having joined GH, draw EF parallel to it through the point E. and becaufe GH is parallel to EF one of the fides of the triangle DEF, DG is to GE, as DH to HFb. but DG is equal E to A, GE to B, and DH to C; therefore as A is to B, fo is C to HF. Wherefore to the three given ftraight lines A, B, C a fourth proportional HF is found. Which was to be done. O PRO P. XIII. PRO B. To find a mean proportional between two given straight lines. Let AB, BC be the two given ftraight lines; it is required to find a mean proportional between them. Place AB, BC in a straight line, and upon AC describe the femi circle ADC, and from the point B draw BD at right angles to AC, and join AD, DC. Because the angle ADC in a femicircle is a right angle, and becaufe in the right angled triangle D ADC, DB is drawn from the right angle perpendicular to the base, DB is a mean proportional between AB, BC the fegments of the bafes. therfore between the two given ftraight lines AB, BC, a mean c. Cor. 96. proportional DB is found. Which was to be done. Book VI. PROP. XIV. THEOR. EQUAL parallelograms which have one angle of the one equal to one angle of the other, have their fides about the equal angles reciprocally proportional: and parallelograms that have one angle of the one equal to one angle of the other, and their fides 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 fides DB, BE be placed in the fame straight line; a. 14. 1. wherefore alío FB, BG are in one straight line. the fides 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 FE angles are reciprocally proportional. But let the fides about the equal angles be reciprocally proporti onal, viz. as DB to BE, fo GB to BF; the parallelogram AB is equal to the parallelogram BC. Because as DB is to BE, fo GB to BF; and as DB to BE, so is the parallelogram AB to the parallelogram FE; and as GB to BF, fo is parallelogram BC to parallelogram FE; therefore as AB to FE, fo BC to FE. wherefore the parallelogram AB is equal to the parallelogram BC. Therefore equal parallelograms, &c. Q. E. D. e PROP. PRO P. XV. THE OR. EQUAL 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 fides 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 fides about the equal angles of the triangles are reciprocally proportional; that is, CA is to AD, as EA to AB. B D Book VI. b. 7. 9. Let the triangles be placed fo that their fides CA, AD be in one ftraight line; wherefore also EA and AB are in one straight line; a. 14. 1. and join BD. Because the 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 EAD to triangle DAB b. but as triangle CAB to triangle BAD, fo is the bafe CA to AD; and as triangle EAD to triangle DAB, fo is the base EA to AB; as therefore CA to AD, fo is EA to AB. wherefore the fides of the triangles ABC, ADE about the equal angles are reciprocally proportional. C E But let the fides 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. C. I. 6. E. 11. §. Having joined BD as before, becaufe as CA to AD, fo is EA to AB; and as CA to AD, fo is triangle BAC to triangle BAD; and as EA to AB, fo triangle EAD to triangle BAD; therefore as triangle BAC to triangle BAD, fo is triangle EAD to triangle BAD ; that is, the triangles BAC, EAD have the fame ratio to the triangle BAD. wherefore the triangle ABC is equal to the triangle ADE. e. p. 3. Therefore equal triangles, &c. Q. E. D. e PROP. Book VI. a. II. I. IF F four ftraight 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 ftraight lines are proportionals. Let the four straight lines AB, CD, E, F be proportionals, viż. as AB to CD, fo E to F; the rectangle contained by AB, F is equal to the rectangle contained by CD, E. From the points A, C drawa AG, CH at right angles to AB, CD; and make AG equal to F, and CH equal to E, and complete the pårallelograms BG, DH. because as AB to CD, fo is E to F; and that b. 7. 5. E is equal to CH, and F to AG; AB is to CD, as CH to AG. therefore the fides 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 e. 14. 6. to one another; therefore the parallelogram BC is equal to the pa rallelogram DH. and the pa And if the rectangle contained by the ftraight lines AB, F be equal to that which is contained by CD, E; thefe 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, beaufe AG is equal to F; and the rectangle DH by CD, E, because CH is equal to É; therefore the parallelogram EG is equal to the parallelogram DH; and they are equiangular. but the fides about the the equal angles of equal parallelograms are reciprocally proportio- Book VI. nal. 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. Wherefore 14. if four, &c. Q. E. D. IF PROP. XVII. THEOR. F three ftraight lines be proportionals, the rectangle contained by the extremes is equal to the fquare of the mean and if the rectangle contained by the extremes be equal to the fquare of the mean, the three straight lines are proportionals. Let the three straight lines A, B, C be proportionals, viz. as A to B, fo B to C; the rectangle contained by A, C is equal to the fquare of B. Take D equal to B; and because as A to B, fo B to C, and that a B is equal to D; A is to B, as D to C. but if four straight a. 7. 5. lines be proportionals, the rectangle contained A is the fquare of B; becaufe B is equal to D. therefore the rectangle contained by A, C is equal to the fquare of B. And if the rectangle contained by A, C be equal to the fquare of B; A is to B, as B to C. The fame conftruction being made, becaufe the rectangle contained by A, C is equal to the fquare of B, and the fquare 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 ftraight lines are proportionals. therefore A is to B, as D to C; but B is equal to D; |