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Book'vi.

PROP. XXI. THEOR.

RECTILINEAL figures which are similar to the same rectilineal figure, are also similar to one another.

Let each of the rectilineal figures, A, B be similar to the rectilineal figure C: the figure A is similar to the figure B.

Because Å is similar to C, they are equiangular, and also have their sides about the equal angles proportionals 2. Again, & 1. Det. 6: because B is similar to C, they are equiangular, and have their sides about the equal angles propor

B tionals a ; . therefore the figures A, B are each of them equiangular to C, and have the sides about the equal angles of each of them and of C proportionals. Wherefore the rectilineal figures A and B are equiangular", and b 1. Ax. 1. have their sides about the equal angles proportionals c. There-c 11. 5: fore A is similar to B. Q. E. D.

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PROP. XXII. THEOR.

IF four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals; and if the similar rectilineal figures similarly described upon four straight lines be proportionals, those straight lines shall be proportionals.

Let the four straight lines AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH, and upon AB,CD let the similar rectilineal figures KAB, LCD be similarly described ; and upon EF, GH the similar rectilineal figures MF, NH in like manner: the rectilineal figure KAB is to LCD, as MF to NH.

To AB, CD take a third proportional a X; and to EF, GH 2 11.6. a third proportional 0 : and because AB is to CD, as EF to b 11. 5. GH, and that CD is b to X, as GH to 0; wherefore, er equali, c 22. 5. as AB to X, so EF to 0: but aş AB to X, so is d the rectilineal a ? Cor. 20.

Book VI. KAB to the rectilineal LCI), and as EF to O, so is d the rection lineal MF to the rectilineal NH; therefore, as KAB to LCD,

so bis MF to NH. 20. 6.

And if the rectilineal KAB be to LCD, as MF to NH; the b 11. 5. straight line AB is tò CD, as EF, to GH. e 12. 6.

Make e as AB to CD, so EF to PR, and upon PR describe! f 18. 6. the rectilineal figure SR simitar and similarly situated to either

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of the figures MF, NH: then, because as AB to CD, soris
EF to PR, and that upon AB, CD are described the similar
and similarly situated rectilineals KAB, LCD, and upon EF,
PR, in like manner, the similar rectilineals MF, SR; KAB,
is toʻLCD, as MF to SR; but, by the hypothesis, KAB is to
LCD, as MF to NH :and therefore the rectilineal MF having
the same ratio to each of the two NH, SR, these are equal
to one another: they are also similar, and similarly situated;
therefore GH is equal to PR : and because as AB to CD, so is
EF to PR, and that PR is equal to GH; AB is to CD, as EF
10 GH. If, therefore, four straight lines, &c.Q. E. D. ;

8 9. 5.

PROP. XXIII. THEOR.

See Note.

EQUÍANGULAR Parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG: the ratio of the parallelogram AC to the parallelogram CF is the same with the ratio which is compounded of the ratios of their sides.

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Let BC, CG, be placed in a straight line; therefore DC and Book VI. CF are also in a straight line a ; and complete the parallelogram DG; and, taking any straight line K, makebas BC to CG, a 14, 1. so K to L, and as DC to CE, so makeb L to M: therefore b 12. 6. the ratios of K to L, and L to M, are the same with the ratios of the sides, viz. of BC to CG, and DC to ÇE. But the ra-:tio of K to M is that which is said to be compoundedc of the c A. def. 5. ratios of K to 1., and I to M; wherefore also K has to M the ratio compounded of the ratios of the

A
sides; and because as BC to CG, so is

DH
the parallelogram ACto the parallelo-
gram CHd; but as BC to CG, so is K
to L; therefore K ise to L, as the pa-.

G

e 11. 5. rallelogram AC to the parallelogram B

C CH: again, because as DC to CE, so is the parallelogram CH to the parallelogram CF; but as DC to CE, so is L to M; wherefore Lise to M, as the parallelogram CH to the parallelogram CF: therefore, since it has been K L M E proved, that as K to L, so is the parallelogram Ac to the pa. rallelogram CH; and as L to M, so the parallelogram CH to the parallelogram CF; ex æqualis, K is to M, as the paral-f 22. 5. lelogram AC to the parallelogram CF: but K has to M the ratio which is compounded of the ratios of the sides; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides. Wherefore, equiangular parallelograms, &c. Q. E. D.

PROP. XXIV. THEOR.

See Note.

THE parallelograms about the diameter of any parallelogram, are similar to the whole, and to one another,

Let ABCD be a parallelogram, of which the diameter is AC; and EG, HK the parallelograms about the diameter: the parallelograms EG, HK are similar both to the whole parallelogram ABCD, and to one another,

Because DC, GF are parallels, the angle ADC is equal a to the angle AGF: for the same reason, because BC, EF are pa

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Book VI. ralleis, the, angle ABC is equal to the angle AEF: and each

of the angles BCD, EFG is equal to the opposite angle DAB, b 34. 1. and therefore are equal to one another; wherefore the paral

lelograms ABCD, AEFG are equiangular : and because the angle ABC is equal to the angle AEF, and the angle BAC

common to the two triangles BAC, EAF, they are equiangu• 4. 6. lar/to one another; therefore cas-AB A

E

В. to BC, so is AE to EF: and because the opposite sides of parallelograms

F d. 5. are equal to one anotherb, AB igd to G

AD, as AE to AG; and DC to CB,
as GF to FE; and also CD to DA,
as FG to GA: therefore the sides of
the parallelograms ABCD, AEFG
about the equal angles are proportion.

D K

С ¢ 1. def. 6. als'; and they are tlierefore similar to one another e: for the

same reason, the parallelogram ABCD is similar to the parallelogram FHCK. Wherefore each of the parallelograms GE, KH is similar to DB: but rectilinçal figures' which are

similar to the same rectilineal figure, are also similar to one £21. 6. anotherf; therefore the parallelogram GE is similar to KH.

Wherefore, the parallelograms, &c. Q. E. D.

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PROP. XXV. PROB.

See Note. TO describe a rectilineal figure which shall be si

milar to one, and equal to another given rectilineal igure.

Let ABC be the given rectilineal figure, to which the figure to be described is required to be similar, and D that to which it must be equal. It is required to describe a rectilineal figure

similar to ABC, and equal to D. a Cor. 45. 1. Upon the straight line BC describe the parallelogram BE

equal to the figure ABC; also upon CE describe a the parallelogram CM equal to D, and having the angle FCE equal to the angle CBI:: therefore BC and CF are in a straight

lipéb, as also LE and EM: between BC and CF finde a mean b 14. 1.

proportional GH, and upon GH described the rectilineal figure c 13. 6. d 18. 6.

KGH similar and similarly situated to the figúre ABC: and e 2. Cor. 20. because BC is to GH as GH to CF, and if three straight

6. Lines be proportionals, as the first is to the third, so ise the

$ 29, 1.

figure upon the first to the similar and similarly described figure Book VI.
upon the second; therefore as BC to CF, so is the rectilineal
figure ABC to KGH: but as BC to CF, so isf the parallelo- f 1.6.
gram BE to the parallelogram EF : therefore as the rectili.
neal figure ABC is to KGH, so is the parallelogram BE to
Che parallelogram EF&: and the rectilinoal figure ABC is 11. 5.

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equal to the parallelogram BE; therefore the rectilineal
fagure KGH is equal h to the parallelogram EF : but EF is h 14. 55.
equal to the figure D; wherefore also KGH is equal to D;
and it is similar to ABC. Therefore the rectilineal figure
KGH has been described similar to the figure ABC, and cqual
to D. Which was to be done.

PROP. XXVI. THEOR,

IF two similar parallelograms have a common an. gle, and be similarly situated; they are about the same diameter.

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Lot the parallelograms ABCD, AEFG be similar and simi-
larly situated, and have the angle DAB common : ABCD and
AEFG are about the same diameter.

For, if not, let, if possible, the A G
parallelogram BD have its da."
meter AHC in a different straight

K

H line from AF the diameter of the

E
parallelogram EG, and let GF

F
meet AHĆ in H; and through
H draw HK parallel to AD or
BC: therefore the parallele-
grams ABCD, AKHG being

B
about the same diameter, they
are similar to one another a : wherefore as DA 10 AB, so is a 24. a
GA to AK: but because ABCD and AEFG ad similar pa- b 1 def. 6.'

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