Book VI. Wherefore the parallelogram AB is equale to the parallelo. gram, BC. Therefore equal parallelograms, &c. Q.E.D.

e 9. 5.

a 14. 1.

b 11.5.

EQUAL

PROP. XV. THEOR.

QUAL triangles which have one angle of the one equal to one angle of the other, have their fides about the equal angles reciprocally propor tional: 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.

B

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.

C

A

E

Let the triangles be placed fo that their fides CA, AD be in one straight line; wherefore alfo EA and AB are in one ftraight line; join BD. Because the triangle ABC is equal to the triangle ADE, and ABD is another triangle; therefore, triangle CAB: triangle BAD: : triangle EAD: triangle BAD; but CAB: BAD :: CA: AD, and EAD: BAD: EA: AB; therefore CA: AD: EA: ABd; wherefore the fides of the triangles ABC, ADE about the equal angles are reciprocally proportional.

But let the fides of the triangles ABC, ADE about the equal angles be reciprocally proportional, viz. CA to AD, as

EA

EA to AB; the triangle ABC is equal to the triangle Book VI. ADE.

Having joined BD as before; because CA: AD :: EA: AB;

and fince CA: AD :: triangle ABC triangle BADc; c 1. 6. and alfo EA: AB triangle EAD: triangle BADc; d11. 5. therefore, triangle ABC: triangle BAD: triangle EAD: triangle BAD; that is, the triangles ABC, EAD have the fame ratio to the triangle BAD: wherefore the triangle ABC is equal e to the triangle EAD. Therefore equal triangles, e 9. 5. &c. Q.E. D.

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, viz. as AB to CD, so E to F; the rectangle contained by ÁB, F is equal to the rectangle contained by CD, E.

a II. I.

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 complete the parallelograms BG, DH. Because AB: CD: E: F; and fince E CH, and FAG, AB: CDb:: CH: AG: therefore the fides of the paral- b 7.5. lelograms BG, DH about the equal angles are reciprocally proportional; but parallelograms which have their fides about equal angles reeiprocally proportional, are equal to one another; therefore the parallelogram BG is equal to c 14. 6.

the

E

Hr

Book VI. the parallelogram DH: and the parallelogram BG is contained by the straight lines AB, F; becaufe AG is equal to F; and the pa- F rallelogram DH is contained by CD and E, because CH is equal to E: therefore the rectangle contained by the ftraight lines AB, F is equal to that which is contained by CD and E.

G

A

B C

D

And if the rectangle contained by the ftraight 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 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 14. 6. equal parallelograms are reciprocally proportional d: wherefore AB CD:: CH: AG; but CHE, and AG F; therefore AB: CD:: E:F. Wherefore, if four, &c. Q. E. D.

IF

PROP. XVII. THEOR.

F three straight 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 ftraight lines are proportionals.

Let the three ftraight lines A, B, C be proportionals, viz. as A to B, fo B to C; the rectangle contained by A, C is cqual to the fquare of B.

Take

A

Take D equal to B; and because as A to B, fo B to C, and that B is equal to D; A is a to B, as D to C: but if four ftraight lines be proportionals, the rectangle contained by the extremes is equal to that which is contained by the means b: therefore the rectangle A.C the rectangle B.D; but the rectangle B.D is equal to the fquare of B, because B=D; therefore the rectangle A.C is equal to the fquare of B.

B

D

C

And if the rectangle contained by A, C be equal to the fquare of B; A: B,:: B: C.

The fame construction being made, because 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 b: therefore, AB:: D: C, but BD; wherefore A: B:: B: C. Therefore, if three straight lines, &c. Q.E.D.

Book VI.

a 7.5.

b 16. 6.

PROP. XVIII. PROB.

[PON a given ftraight line to defcribe a rectilineal figure fimilar, and fimilarly fituated to a given rectilineal figure.

Let AB be the given ftraight line, and CDEF the given rectilineal figure of four fides; it is required upon the given ftraight line AB to describe a rectilineal figure fimilar, and fimilarly fituated to CDEF.

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 a 23. I. angle ABG equal to the angle CDF; therefore the remaining angle CD is equal to the remaining angle AGB b 32. I Wherefore the triangle FCD is equiangular to the triangle GAB: Again, at the points G, B in the ftraight line GB

make

a 23. I.

Book VI. make the angle BGH equal to the angle DFE, and the angle GBH equal to FDE; therefore the remaining 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

H

b 4. 6.

C 22. 5.

for the fame reason, the angle ABH is equal to the angle CDE; alfo 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 fides about the equal angles proportionals: for the triangles GAB, FCD being equiangular,

BA: AG:: DC: CF b; for the fame reason, AG: GB:: CF: FD; and because of the equiangular triangles, BGH, DFE, GB: GH:: FD: FE; therefore, ex aequali c, AG: GH:: CF: FE.

In the fame manner, it may be proved, that
AB: BH: CD: DE.

Also b,

GH: HB:: FE: ED. Wherefore, because the rectilineal figures ABHG, CDEF are equiangular, and have their fides about the equal angles proportionals, they are d 1. def. 6. fimilar to one another d.

Next, Let it be required to defcribe upon a given ftraight line AB, a rectilineal figure fimilar, and fimilarly fituated to the rectilineal figure CDKEF.

Join DE, and upon the given ftraight line AB defcribe the rectilineal figure ÁBHG fimilar, and fimilarly fituated to the quadrilateral figure CDEF, by the former cafe; and at the points B, H in the ftraight 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 becaufe the figures ABHG CDEF are fimilar, 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 fame

reafon