GEOMETRY. 



the other sides, or the other sides produced propor- 

 tionally, and conversely. 



Let ABC be a triangle, and let DE be drawn 

 parallel to the side BC, meeting AB and AC, or 



or 



those sides produced in D and E ; it is required 

 to prove that 



AD_AE 



DB = EC 



Join BE and CD. Then since the two triangles 

 BDE, ADE have the same altitude, they are to 

 each other as their bases, 



BDE = BD 



ADE DA' 



DEC CE 

 Similarly, 5^ - gy 



But DEC and BDE are equal, being on the 

 same base, and between the same parallels ; 

 . BD _ CE 

 '' DA EA' 

 AD _ AE 

 DB EC' 

 BD DA CE EA 



or 



or 



DA 



or 



_. , ., . . 



Conversely, if TVA = ifr 



BA 

 AD 

 BD 

 DA 



EA 

 CA 

 AE' 

 CE 



' 



to prove that DE is parallel to BC. 

 parallel to BC, draw DF parallel 

 to BC; 



BD CF 

 411611 = " 



If DE is not 



But 



BD CE 



. CF 

 ' ' FA 



CE 

 EA' 



which is absurd ; therefore DE is parallel to BC. 



Definition. Rectilineal figures are similar when 

 they are equiangular, and have also the sides 

 about the equal angles proportional 



SIMILARITY OF TRIANGLES. 

 PROP. XXXI. Two triangles are similar : 



(i.) If two angles of the one be equal to two 

 angles of the other. 



(2.) If the sides about two angles in the one be 

 proportional to the sides about two angles in the 

 other. 



(3.) If an angle of the one be equal to an angle 

 of the other, and the sides about the same angles 

 .be proportional. 



(4.) If an angle of the one be equal to an angle 

 of the other, and the sides about two other angles 

 be proportional, the remaining angles being each 

 either less or greater than a right angle. 



Let ABC and 

 abc be two tri- 

 angles ; it is re- 

 quired to prove 

 that they are simi- 

 lar : 



(i.) If angle A 

 equal to angle a, 



angle B equal to angle b ; it is required to prove 

 that the triangles are similar that is, that they are 

 equiangular, and have the sides about the equal 

 angles proportional. They are equiangular, for 

 the remaining angle C in the one is equal to the 

 remaining angle c in the other ; they have also 

 the sides about the equal angles proportional. 

 For if the triangle abc be applied to ABC, so that 



a may lie on A, and ab on AB, then ac shall lie on 

 AC, since angle A is equal to angle a. 



Let ADE be the new position of abc, then since 

 angle ADE is equal to angle ABC, DE (Prop. IX.) 

 is parallel to BC ; hence 



AB AC 

 AD "" AE' 



or -2- = . 



ab ac 



Similarly, by applying the triangle abc to ABC, 

 so that b lies on B, and ba on BA, we have 



AB BC 

 ab be ' 



. AB = AC = BC 

 ab ac be' 



Therefore the sides about the equal angles are 

 proportional, and it was proved that the triangles 

 are equiangular ; hence they are similar. 



AB BC 



and 



ab 



ac 



it is required to prove that the triangles are similar. 



At the point b in 



the line be make 



the angle cbd 



equal to the angle 



CBA, and at the 



point c in the line 



be make the angle 



bed equal to the angle BCA. Then, by the last case, 



AB BC 

 db be' 



But 



BC , AB , . ., . 

 -j-- equals r by hypothesis ; 



AB . AB 



' -M C< l UalS -^T ' 



Similarly, 



/. db equals ab. 

 dc equals ac ; 



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