BOOK vi. PKOP. xxr. xxn.] MATHEMATICS. PLANE GEOMETRY. 539 



10 v. sequently,* A B E : F G L : : polygon ABODE: 

 polygon FGHKL; but ABE:FGL:: A B 2 : FG 2 ,t 



+ 19 VI. . . (2V.) the polygons are to one another as the 

 squares of {heir homologous sides A B, F G ; similar poly- 

 gons, ire. Q. E. D. 



COR. It follows from this and the corollary to Prop. 

 XIX. , that if three straight lines be proportionals, the 

 first is to the third as any rectilineal figure upon the 

 first, to a similar and similarly described figure upon 

 the second. 



PROPOSITION XXI THEOREM. 



Rectilineal figures (A B) which are similar to the same rec- 

 tilineal figure (C) are also similar to one another. 



Because A is similar to C, they are equiangular, and 

 Def i vi. have also their sides about the equal angles 

 proportionals. * 



Again, because B is similar to C, they also are equi- 

 angular, and have their sides about the equal angles pro- 

 portionals, .". A, B are each of them equiangular to C, 

 and have their sides about the equal angles of each of 

 them and of C proportionals ; .'. A, C are equiangular, 

 and have their sides about the equal angles propor- 



+ 2 v. tionals ;t '. A, B are similar figures ; . . 

 rectilineal figures, tc. Q. E. D. 



PROPOSITION XXII. THEOREM. 

 If four straight lines (A B, C D, E F, G H) are propor- 

 tionals, the similar rectilineal figures, similarly described 

 upon them, are also proportionals : and if the similar 

 and similarly described rectilineal figures upon futtr 

 ttraiyht lines are proportionals, the straight lines them- 

 selves also are proportionals, 



Draw A K, C L, making any equal angles with A B, 

 C I ) ; make also A K = E F, and C L = G H ; draw 

 KB, LD. 



By hypothesis and construction A B : A K : : C D : 

 C L ; and the angles A, C are equal ; . . K A B, L C D 



6 vi. are equiangular ;* '.' 



+ 19 vi. A B 1 : f D : : triangle K A B : triangle LCD 1 1 

 A K. 1 : C L 1 : : triangle K A B : triangle LCD! 



iv. .-. AB :CL 2 :: AK^CL*.* 



But a polygon on A B : similar polygon on CD:: 

 A B 2 : C D 1 ; and poly- 

 gon on A K : similar E t 

 polygon on CL: : AK 1 : ,. q u 

 C L 2 ; .'. polygon on / \ I, 

 A B : aim. polygon on 

 CD:: polygon on E F : 

 sim. polygon on G H. 



Again, let the last A T> 



proportion have place, then AB:CD::EF:GH. 



Make equal angles at A and C, as before, as also A K = 



EF; and as AB is to C D. so make AK to CL;t 



T 11 VI. then the triangles K A B, LCD are equian- 



e vi gular ;* and .'., as proved above, AB 1 : C D 2 : : 

 A K 1 : C L 2 ; but A K = E F, /. A B 2 : C D 2 : : E F 2 : 

 CIA But by hyp., the polygons on AB, CD, E F, 

 G H are proportionals, and therefore, since similar poly- 



t 20 VI gons are as the squares of their like sidesf A B': 



9V.Cor.l. CD 2 : ^F 2 : G H 2 ; .'. CL = GH.* But 



by construction, the four lines A B, C D, A K, C L, are 



proportionals ; .'. the four lines A B, C D, E F, GH are 



proportional* ; .', if four straight lines, <feo. Q. E. D. 



PROPOSITION XXIIL THEOREM. 

 Equiangular parallelograms (A C, D F) are to each other 



as the rectangles of their containing sides (A B - B C and 



DEEF). ' 



Draw AG, DH, perpendiculars to BC, EF. Then 

 because parallelograms on the same base and between 

 the same parallels are equal, A C = A G'B C, and D F 

 = D H-E F. Also the rectangles A G B C, A B B C 

 having the same alti- 4 



tude B C, are to each 

 oth.T as their bases, 

 A G, A B. In like 

 manner, the rectan- 

 gles D H-E F, D E- 



1 vi. E F are to each other as their bases D H, D E. * 

 | But the triangles A B G, D E H having the angles at B 

 : and G respectively equal to those at E and H, are equi- 



+ 4 vi. angular; .'. AG :AB :: DH: DE.t ' AG- 

 B C : AB'B C : : D H-E F : D E-E F, and alternately* 



liv. AG-BC:DH-EF:: AB-BC : DE-E F; that 

 is, the parallelograms A C, D F are to each other as the 

 rectangles A B'B C, D E'E F of their ctmtaining sides ; .'. 

 equiangular parallelograms, <tc. Q. E. D. 



PROPOSITION XXIV. THEOREM. 



Parallelograms (E G, H K) about the diagonal (A C) of a 

 parallelogram (D B) are similar to the whole, and to 

 one another. 



Because D C, G F are parallels, the angle A D C= 

 + 291. AGFj-f and because BC, E F are parallels, 

 A B C = A E F ; and each of the an- 

 gles BCD, EFG is equal to the 

 opposite angle DAB; .'. B C D = 

 EFG; .'. the parallelograms B D, 

 E G are equiangular. And because 

 the angle ABC = AEF, and that 

 B A C is common to the two tri- 

 angles BAG, E A F, .'. they are 

 equiangular ; /. A B : B C : : A E : 



4 vi. EF;* that is, the opposite sides of parallolo- 

 A D : A E A G \ .-. the sides 

 C B : G F F E of the equi- 

 D A : F G G A ) angular pa- 







grams being equal, A B 

 DC 

 CD 



M 



rallelograms B D, EG, about the equal angles, are 

 proportionals : the parallelograms are therefore similar. 

 For like reasons, the parallelograms B D, H K are 

 similar ; . . each of the parallelograms E G, H K is 

 similar to B D : they are therefore similar to each otter ;* 



21 vi. . ' . parallelograms, <tc. Q. E. D. 



PROPOSITION XXV. PROBLEM. 



To describe a rectilineal figure which shall be similar to ont 



(P) and equal to another given rectilineal figure (D). 



Upon B C, a side of the given figure, describe the 

 4- 4s I. Cor. parallelogram B E = P ;t and upon CE describe 

 the parallelogram C M = D, and having the angle F C E 

 = CBL; then LBO 

 -f ECB = two riyht 

 angles, .'. FCE + 

 E C B = two right 

 angles, .-. B C, C F 

 are in a straight 



H i. line ; * so 8 

 also for like reasons 

 are L E, EM. Be- 

 tween B C, C F find 



1 13 vi. a mean proportional G H ; t and upon it de- 

 scribe the rectilineal figure Q, similar and similarly 



is vi. situated to the figure P,* Q shall be the figure 

 required. 



Because B C : G H : : GH : OF, . -. BC : C F : : P : 



+ 20 vi. cor. Q ; + but B C : C F : : B E : E F ; . -. P : Q : : 



BE : EF; and P = B E, . -. Q = E F ; but EF = D, 



. . Q = D, and it is similar to P. Which was to be done. 



PROPOSITION XX VI. THEOREM. 



If two similar parallelograms (B D, EG) have a common 

 angle (at A) and be similarly situated, they are about the 

 same diagonal. 

 For if not, let, if possible, the parallelogram B D have 



its diagonal A H C in a different straight line from A F, 



the diagonal of E G ; and let G F, or G F prolonged, 



meet A H C in H, and draw H K 



parallel to D A, or C B. Then 



B D, K G being about the same 



diagonal are similar, . . DA: 



A B : : G A : A K ; but B D, EG 



Hyp. are similar ; * . ' . DA: 

 AB::GA:AE;.'. GA:AE:: 

 G A : A K ; . -. A K = A K which 

 is impossible; .'. B D, KG are 



not about the same diagonal, that is, the diagonal of B D 



