CHAMBERS'S INFORMATION FOR THE PEOPLE. 



then the two triangles dbc and abc are equiangular, 

 but ABC and dbc are equiangular ; 



.'. ABC and abc are also equiangular, 

 hence they are similar. 



(3.) If angle A be equal to angle a, and the 

 sides about these angles be proportional namely, 



AB = AC . 



ab ac ' 



it is required to prove that the triangles are similar. 

 For if the tri- 



angle abc be A 



applied to 

 ABC, so that 

 a lies on A, 



and so that ab / _ \ 



lies along AB, & c 



then will ac lie 



along AC, since the angles at A and a are equal. 

 Let ADE be the position of abc. Then since 



AB AC 



AD "* AE' 



DE must be parallel to BC ; 

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

 and AED to ACB. 



But ADE is equal to abc, and AED to acb, 

 therefore the angle abc is equal to ABC, and acb 

 to ACB ; hence the two triangles ABC, abc are 

 similar. 



(4.) If angle A be equal to angle a, and the 

 sides about angles B and b be proportional 

 namely, 



AB_ BC 



ab ~ be' 



the angles at C and c being each either less or 

 greater than a 

 right angle ; it 

 is required to 

 prove that the 

 triangles are 

 similar. If the 

 angles at B 

 and b are equal, then the triangles are equiangular, 

 and consequently similar. But if angle B be not 

 equal to angle b, at the point B in AB make the 

 angle ABD equal to the angle abc. Then the two 

 triangles ABD, abc are equiangular ; 

 . AB = BD 

 ab be' 



But 



- 

 ab ~ be 



. BD _ BC 



'* be ~ be ' 



.'. BD = BC 



Then, first, if we suppose the angles at C and c 

 both acute ; 

 since BD = BC, 

 angle BCD = angle BDC (Prop. IV.) ; 

 and since angle BCD is acute, angle BDC is also 

 acute ; therefore angle BDA must be obtuse, but 

 BDA is equal to bca, which is acute ; therefore 

 BDA is both acute and obtuse, which is absurd ; 

 therefore the angle at B is equal to the angle at b, 

 and the triangles are similar. In like manner, if 

 the angles at C and c be supposed both obtuse, it 

 may be shewn, that the triangles are similar. 



PROP. XXXII. Two triangles, which have an 

 angle of the one equal to an angle of the other, are 



624 



to each other in the ratio compounded of the ratios 

 of the sides about the equal angles. 



Let ABC and 



abc be two tri- A 



angles which 

 have the angle 

 A equal to the 

 angle a ; it is 

 required to prove 

 that 



ABC = AB AC 

 abc ab Ac ' 



Let the triangle abc be applied to ABC, so that 

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

 lie on AC. Join B^. Then we have 



ABC AC 



= -j- , since they have the same altitudes, 



AEc AB 



Abc M' 



and 



Multiplying these ratios, we have 



AB 



Me 



AB AC 

 M X Ac ' 



But Abc =abc, Ab = ab, and Ac = ac. 



ABC 



abc 



AB AC 

 ab ac' 



Cor. i. If the two triangles ABC, abc are simi- 

 lar, we have 



AB = AC. 

 ab ac ' 

 . ABC = (AB) 3 

 " abc ~~ (ab)'' 



It follows that similar triangles are to each other 

 as the squares upon their corresponding sides. 



Cor. 2. If the two triangles ABC and abc are 

 equal, we have 



AB- AC = ab- ac, 



AB ac 

 or r = --= ; 



hence it follows that in equal triangles, if any 

 angle of the one be equal to an angle of the other, 

 the sides about the equal angles are reciprocally 

 proportional. It is also evident that if the sides 

 are reciprocally proportional, the triangles are 

 equal. 



Cor. 3. Hence it follows that equiangular 

 parallelograms have to one another the ratio* 

 which is compounded of the ratios of their sides. 



PROP. XXXIII. Similar rectilineal 

 may be divided into the same number of simil 

 triangles, and they are to each other as the square 

 upon their corresponding sides. 



Let ABCDE, abcde be two similar rectilineal 

 figures, then they 

 may be divided into 

 the same number of 

 similar triangles, and 

 they are to each 

 other as the squares 

 on the corresponding 

 sides. Join AC and 

 ac. Then the two 



triangles ABC, abc are similar, since angle 

 equals angle b, 



AB . BC 



and j- equals -r . 



ab ^ be 



