grams ab, bc about the equal angles, are reciprocally proportional; that

is, db is to be as gb to bf.

f

Complete the parallelogram fe; and because the parallelogram ab is equal to bc, and that fe is another parallelogram, a b is to fe as bc to fe (v. 7): a but as ab to fe, so is the base db to be (vi. 1) and as bc to fe, so is the base of gb to bf; therefore, as db to be, so is gb to bf (v. 11). Wherefore, the sides of the parallelograms ab, bc about their equal angles are reciprocally proportional.

e

d

b

g

с

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; therefore as a b to fe, so is bc to fe (v. 11): wherefore the parallelogram a b is equal (v. 9) to the parallelogram bc. Therefore equal parallelograms, &c. Q. E. D.

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 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.

LET abc, a de 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 a d, as ea to a b.

b

d

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 (i. 14); 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, so is the triangle ead to triangle dab (v. 7): but as triangle cab to triangle bad, so is the base ca to ad (vi. 1); and as triangle ead to triangle da b, so is the base e a to a b (vi. 1): as therefore ca to a d, so is e a to a b (v. 11); wherefore the sides of the triangles a b c, ade about the equal angles are reciprocally proportional.

C

a

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 a d, so is triangle abc to triangle bad (vi. 1); and as e a to a b, so is triangle e ad to triangle bad (vi. 1); therefore (v. 11) as

triangle bac to triangle bad, so is triangle e ad to triangle bad: that is, the triangles bac, ead have the same ratio to the triangle bad: wherefore the triangle abc is equal (v. 9) to the triangle ade. Therefore equal triangles, &c. Q. E. D.

PROPOSITION XVI.-THEOREM.

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, c d, e, f, be proportionals, viz. as ab to cd, so e to f; the rectangle contained by a b, f is equal to the rectangle contained by cd, e.

From the points a, c, draw (i. 11) ag, ch at right angles to a b, cd; and make ag equal to f, and ch equal to e, and complete the parellelograms bg, dh; because, as a b to cd, so is e to f; and that e is equal to ch, and f to a g; ab is (v. 7) to cd as ch to a g. Therefore the sides of the parallelograms bg, dh about the equal angles are reciprocally proportional; but parallelograms which have their sides about equal angles

e

f

g

h

reciprocally proportional, are equal to one another (vi. 14); therefore the parallelogram bg is equal to the parallelogram dh and the parallelogram bg is contained by the straight lines ab, f; because ag is equal to f; and the parallelogram dh is contained by cd and e; because ch is equal to e: therefore the rectangle contained by the b c d straight lines ab, f is equal to that which is contained by cd and e. And if the rectangle contained by the straight lines a b, 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

a

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 a b, f, because a g 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 sides about the equal angles of equal parallelograms are reciprocally proportional (vi. 14): wherefore, as ab to cd, so is ch to ag; and ch is equal to e, and ag to f: as therefore ab is to cd, so is e to f. Wherefore, if four, &c. Q. E. D.

PROPOSITION XVII.—THEOREM.

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.

:

a

b

d

Take d equal to b; and because as a to b so b to c, and that bis equal to d a is (v. 7) to b as d to c: but if four straight lines be proportionals, the rectangle contained by the extremes is equal to that which is contained by the means (vi. 16): therefore the rectangle contained by a, c is equal to that contained by b, d: but the rectangle contained by b, dis the square of b; because b is equal to d: therefore the rectangle contained by a, c is equal to the square of b.

a

с

a

b

And if the rectangle contained by a, c be equal to the square of b; a is to b as b is 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 (vi. 16): therefore a is to b as d to c; but b is equal to d; wherefore, as a to b, so b to c: therefore, if three straight lines, &c. Q. E. D.

PROPOSITION XVIII.-THEOREM.

Upon a given straight line to describe a rectilineal figure similar, and similarly situated, to a given rectilineal figure.

LET a b 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 situated, to cdef.

Join df, and at the points a, b in the straight line ab make (i. 23) the angle bag equal to the angle at c, and the angle a bg equal to the angle cdf; therefore the remaining angle cfd is equal to the remaining angle agb (i. 32). Wherefore the triangle fcd is equiangular to the triangle gab: again, at the points g, b in the straight line g b, make (i. 23) the angle bg h 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

K

angle a gb is equal to the angle cfd, and bgh to dfe, the whole angle a gh is equal to the whole cfe: for the same reason, the angle a bh 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 sides about the equal angles proportionals; because the triangles gab, fcd being equiangular, ba is (vi. 4) to a g as dc to cf; and because a g is to gb as cf to fd; and as gb to gh, so, by reason of the equiangular triangles bgh, dfe, is fd to fe; therefore, ex æquali (v. 22), ag is to gh as cfto fe in the same manner it may be proved that a b is to bh as cd to de: and gh is to hb as fe to ed (vi. 4). Wherefore, because the rectilineal figures abhg, cdef are equiangular, and have their sides about the equal angles proportionals, they are similar to one another (vi. def. 1).

:

Next, let it be required to describe upon a given straight line a b, a rectilineal figure similar, and similarly situated, to the rectilineal figure cdkef.

Join de, and upon the given straight line ab describe the rectilineal figure a bhg 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 e dk, and the angle bhl equal to the angle dek; therefore the remaining angle at k is equal to the remaining angle at 1: 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 gh1 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, so is ed to ek (vi. 4); therefore, ex æquali (v. 22) g h is to hl, as fe to ek for the same reason, a b is to bl as cd to dk and blis to lh as (vi. 4) dk to ke, because the triangles blh, dke are equiangular : therefore because the five-sided figures aghlb, cfekd are equiangular, and have their sides about the equal angles proportionals, they are similar to one another; and in the same manner a rectilineal figure of six or more sides may be described upon a given straight line similar to a given rectilineal figure, and so on. Which was to be done.

PROPOSITION XIX.—THEOREM.

Similar triangles are to one another in the duplicate ratio of their

homologous sides.

LET abc, def be similar triangles, having the angle b equal to the angle e, and let a b be to bc, as de to ef, so that the side bc, is homologous to ef (v. def. 12): the triangle abc has to the triangle def, the duplicate ratio of that which bc has to ef

Take bg a third proportional to bc, ef (vi. 11) so that be is to ef as ef to bg, and join ga: then, because, as ab to bc, so de to ef; alternately (v. 16) a b is to de as bc to ef: but as bc to ef, so is ef to bg; therefore (v. 11) as ab to de, so is ef to bg. Wherefore the sides of the triangles abg, def, which are about the equal angles, are reciprocally proportional: but triangles which have the sides about two

equal angles reciprocally proportional are equal to one another (vi. 15) : therefore the triangle a bg is equal to the triangle def: and because as bc is to ef, so is ef to bg; and that if three straight lines be proportionals, the first is said (v. def. 10) to have to the third the duplicate ratio of that which it has to the second; bc therefore has to bg the duplicate, ratio of that which be has to ef: but as bc to bg, so is (vi. 1) the triangle abc to the triangle a bg. Therefore the triangle abc has to the triangle abg the duplicate ratio of that which bc has to ef: but the triangle abg is equal to the triangle def: wherefore also the triangle abc has to the triangle def the duplicate ratio of that which bc has to ef. Therefore similar triangles, &c. Q. E. D.

COR. From this it is manifest, that if three straight lines be proportionals, as the first is to the third, so is any triangle upon the first to a similar and similarly described triangle upon the second.

PROPOSITION XX.-THEOREM.

Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons have to one another the duplicate ratio of that which their homologous sides have.

LET abcde, fghkl be similar polygons, and let ab be the homo