FOUNDED ON ORDINARY GEOMETRY, &c. 



169 



[The equivalent theorem in proportions is 



a : b :: A : B.^ 



>^^ 



Apply one triangle upon the other as in the right-hand diagram, so that the side h meets 

 the hypothenuse A at right angles, and the vertex of the angle opposite 6 meets the vertex of 

 the angle included by A and B. Since the angle GFH is equal to the angle FDE, it is the 

 complement of the angle DFE ; and GFE is therefore a right angle ; and GF is parallel to 

 DE. Now the rectangle under a and B is the double of the triangle GFE ; and the rect- 

 angle under 6 and A is the double of the triangle GFD. But because GF is parallel to DE, 

 the triangle GFE is equal to the triangle GFD. Therefore the rectangle under a and B is 

 equal to the rectangle under A and b. q.e.d. 



Proposition (D). If a, c, and A, C, are homonymous sides of equiangular triangles, the 

 rectangle contained under a, C, will be equal to the rectangle contained under c, A. 



From the angles included by the sides A, C, and a, c, let fall the perpendiculars B, b, 

 upon the third side. The corresponding right-angled triangles thus formed are easily shewn 

 to be equiangular. Hence, by Proposition (C), 



Rectangle under a, B, is equal to rectangle under A, b. 



Again, Rectangle under b, C, is equal to rectangle under B, c. 



Therefore by Proposition (B), 



Rectangle under a, C, is equal to rectangle under A, c. q.e.d. 



Proposition (E). If b, c, and B, C, are homonymous sides including the right angles 

 of two equiangular right-angled triangles, the rectangle contained under 6, C, will be equal 

 to the rectangle contained under c, B. 



This may be considered a case of the last proposition, or it may be treated independently 

 thus. 



Vol. X. Part I. 



22 



