MATHEMATICS PLANE GEOMETRY. [BOOK VLPBOP. T. Tin. 



and in the aame straight line with it ; then the angle 

 , Hyp. BCA-CED.t Add to 

 ach the angle B ; thenBCA + B 

 D + 1! ; !mt the fornu-r t.. 

 are lem than two right angles,* 

 w I. /. CED + B, are less 

 than two right angles, .'. B A, E 1), 

 + AI. ll if produced will meet ;t 

 let them be produced and meet in 

 F. Then because the angle B 

 w >!. DCE.BK is parallel to CD ; 



and because A C B - D E C, A C is parallel to F E ; .'. 

 + *4 I F C is a parallelogram, t .'. A F - C D, and 

 A C - F I) ; and because A C, C D are respectively parallel 

 to F E, B F, sides of the triangle F B E .'. (1 VI. ) 

 B C : C E : : B A : i or since A F , B C : C E : : B A : 

 AFandBC:CE { - CD, and ] CD ; B C : CE: : 



FD:DK FD - AO < AC:DE 

 iv .-. BA:CD::AC:DE. Alternating these 

 three proportions, B C : B A : : CE : CD ; BC:AC: : 

 CE :DE ; B A : AC ::CD :DE. .-.the sides about 

 tht equal angles, .to. Q. E. D. 



COB. In similar triangles (A B C, D E F), the bases 



(BC, EF) are to one 

 anotheras the altitudes 

 (AG, DH). For since 

 the angles B, E are 

 equal, as also the right 

 angles G, H, the tri- 

 ^ angles A B G, D E H 

 are equiangular, . . 

 BC:AB::EF:ED; 



AB :AO 



but 



Def.ivi. .'. BC : AG :: EF: D H ;t hence, alter- 

 + 15 v. Oar. nately, the bases are as the altitudes. 



PROPOSITION V. THEOREM. 



Jf the tides of two triangles (ABC, DBF), about two 

 angles (B,C) of one, and two angles (E, F) of the other, 

 be proportionals, the triangles shall be equiangular ; and 

 the equal angles shall be those which are opposite to 

 homologous sides. 



At the points E,F, in the straight line E F, make the 

 angle F E G =* B, and E F G = C ; then the remaining 

 angle G = A, and . . the triangles A B C, G E F, are equi- 



4 VI. angular, .-.AB:BC::GE:EF;* but (hyp.) 

 + jn. AB:BC::DE:EF;.-.DE:EF::GE:EF;t 



^ .-.DE = GE. Foralikerea- 



/\ son D F = F G. And because 



/\ D in the triangles D E F, G E F, 



/ \ /\ D E = G E, and E F common, 



/ 1 . _ / \ , the two sides D E, E F are 



B C s \ / l equal to the two G E, E F, each 



\7 t each, and D F = OF ; . . 

 81. the angle D E F=G E F,* 

 and the angle D F E = G F E, 



and ED F - E G F. And because D E F = G E F, and 

 t coiut. G E F = B,f . '. D E F = B. For a like rea- 

 aon D FE = C, and . . D = A ; . . the triangle D E F is 

 equiangular to A B C ; . . if the sides, A-c. Q. E. D. 



PROPOSITION VI. THEOREM. 



If two triangles (AB C, D E F) have an anjle (A) of one, 

 equal to an angle (E D F) of the other, and the sides 

 about them proportionals, the triangles shall be equi- 

 angular, and shall have those angles equal which are 

 opjmrite to the homologous sides. 



At the points D, F, in the straight line D F, make the 

 angle F I) G equal to either of the angles A, or E D F ; 

 and the angle D F G equal to C ; then the remaining angle 

 G = B, and . . the triangles 

 A B C, D G F are equian- 

 gular ; . . B A : A C : : G D 

 |0 4VI. :DF; but (hyp) 

 BA: AC: :ED: :DF;.-. 

 ED:D'F::GD:DFjt.-. 

 tsv . ED-GD;.'.in 



the triangles E D F, GD F, 



8 n T r the two sides E D. D F - 



Cout GD, DP, each to each, and theangleEDF - 



GDF;*.-.EF-FGt, and the angle D F G - D F E, 

 + 41. andO -"E;* but DFG -C (const) DFE - 

 41. C j but E D F A (conrt.), . '. E B ; . . the 



triangle* A B C, D E F ore equiangular; .\if two triangles, 



.v c. Q. E. D. 



PROPOSITION VIL THBOBEM. 



// two triangles (A B C, D E F) have an angle (A) of the 

 one, equal to an angle (D) of the other, and the sidet 

 about two (A B C, D E F) proportional* ; then if each 

 of the remaining angles (C, F) be either less, or not less, 

 man a ri'jht angle, the triangles shall be equiangular, 

 and shall have those angles equal, about which the sides 

 are proportionals. 



First let each of the angles C, F bo few than a right 

 angle : the angles ABC and E shall be equal, and . . the 

 angle C to the angle F. For if A B C be not equal to E, 

 one of them, as A B C, must be the greater. At the 

 point B, in A B, make the angle A B G = E ; then be- 

 D (hyp.) the remaining angle A G B must be 

 F ;* .'. the triangles A B G, D E F are equi- 

 angular, .-. A B : B G 

 ::DE:EF. But 

 (hyp.) A B : BC : : 

 DE :EF, .-. AB: 

 BC::AB:BG,.-. 

 BC = BG,tand.'. 



+9V.Cor. l. BGC= 



C B F BCG; but (hyp.) 



B C G is less than a right angle, . . B G C is less than a 

 right angle ; . . A G B must be greater than a right 

 angle : but it was proved that A G B = F ; . . F is 

 greater than a right angle ; but (hyp.) it is less than a 

 right angle ; which is absurd ; . . the angles A B C, E 

 are not unequal, that is, they are equal ; and, as A = D 

 (liyp.). ' .C=F . ' . the triangles ABC,DEF are equiangular. 

 Next, let C, F be each not less than a right angle : the 

 triangles shall also in this case be equiangular. For if 

 A it be denied, then, 



the same construction 

 I, being made, it may be 

 )>n ivtsd, as above, that 

 BC = BG; and that 

 .-. theangleBGC = 

 C : but (hyp.) C is 

 not less than a right 

 angle .-. BG C is not less than a right angle .'. two 

 angles of the triangle B G C are together not less than 

 two right angles : which is impossible ; .'. the triangle 

 ABC, as in the first case, t' equiangular t<> DBF. 

 if two triangles, <fcc. Q. E. D. 



PROPOSITION VIII. THEOREM. 

 In a right-angled triangle (ABC), if a perpendicular 

 (A D) be drawn from the vertex of the right angle to the 

 base, the triangles (A B D, A C D) on each side of it are 

 similar to the whole triangle (A B C) and to one another. 

 Because the angle B A C = A D B, each being a right 

 angle, and that B is common to the 

 two triangles A B C, A B D, .'. the 

 .321. angle ACB = BAD;*. '. 

 the triangles A B C, A B D are equi- 

 + 4 vi. angular, .'.they are similar, t 

 mnd Def.l. In like manner it may be 

 demonstrated that the triangles 

 ABC, A CD are similar; .-.the 

 triangles A B D, A C D being both 

 similar to ABC, are similar to each 

 other ; .' . in a rii/ht-anyled triangle, <fcc. Q. E. D. 



COR. From this it is manifest that the perpendicular 

 from the vertex of the right angle of a right-angled tr- 

 angle to the base, is a mean proportional between the 

 segments of the base ; and also that each side about tho 

 right angle is a mean proportional between the base, and 

 the segment of it adjacent to that aide : for in the tri- 

 4VI. angles B D A, CD A, B D : D A : : D A : D C ;* 

 in the triangles A B C, DB A, BO : B A : : B A : BD ; 

 and in the triangles A B C, A C D, BC:CA ::CA:CD. 



