116 Mk WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULA, 



These formulee, which are remarkable for their symmetry and sim- 

 plicity, suggest various solutions to the problem enunciated in Art. 1. 

 Their evident analogy to the property of a triangle " that the sines of 

 the angles are proportional to the opposite sides", has suggested another 

 form under which they may be put. 



12. The hypothesis and notation of Art. 10. in regard to the tri- 

 angle ABC (fig- 5. No. 1. and 6.) being retained, another triangle A'BC 

 (fig. 5. No. 2.) having remarkable relations with it, may be constructed 

 as follows : 



Let straight lines D'A', D'B', DC meet in a point Z>', the angles 

 A'DB, BBC, ABC being equal to ABB, BBC, ABC respectively. 

 At A' any point in B'A make the angles B'A'B' equal to BBA, and 

 B'A'C equal to BCA, thus forming two triangles B'A'B', B'A'C 

 (fig. 5. No. 2.) similar to BBA, BCA (fig. 5. No. 1). Join BC; be- 

 cause BB : BA = BA' : BB' and BA : BC = BC : BA' ; there- 

 fore, ex (eq. BB : BC ^ B C : BB; hence the triangles BBC, C BB 

 are similar. 



Let the lines and angles in the triangle ABC be expressed by the 

 same letters as are used for the triangle ABC, with the distinction of 

 an accent over such as differ in magnitude, so that 



BC = a, AC = b', AB = c, BA = x', BB' = y', BC = z. 



The angles about B in the two triangles being equal ; viz. t/z = «, 



A A 



x'» = 13, x'y = y. 



13. The similarity of the partial triangles which constitute- the two 

 triangles ABC, A'B'C, besides the equal ratios x : y = y' : xf, hy which 

 they were formed, give us also x : c = y' : c, y : c = x : c ; therefore 



X _ c y 

 ji' ~J'~x' 

 hence the lines in the two triangles have the following properties ; 



xx' = i/y', '— = —, = —'■, ii'id a like result for each pair of triangles ; 



