140 PROCEEDINGS OF THE AMERICAN ACADEMY 



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£g = ^^ = J = n' ; also let C B' = d, C'B = d'. From the 



triangle ABB' we get, by prop. 1 (by using iV, instead of n, to rep- 

 resent the angle A in this triangle, since n represents the angle A 

 in the isosceles triangle ABC),c^=^ a^ -\- a'~ — 2 Naa',{l),or if we 

 put N=nx, we get c^ =^ or -\- a'^ — 2nxaa' ; and in like manner we 

 get from the triangle B'B C, c^= a^ -{• d^+_2n x' a d, (2) ', where 

 for + we must use -\- when B' is not between the points A and C, 

 and — must be used for + when B' is between A and C, as is evi- 

 dent from (1) of prop. 2. Equating the two values of c^, we get, 

 after a slight reduction, a'- — d^ — 2 wxaa' ip 2 na;' ad =0, (3); 

 since 2na = AC, and a'^ — d-= {a' + d) {a' — d) = {a' ± d) X 

 A C (the upper sign being used when B' is not between A and C, 

 and — in the contrary case) ; hence, substituting the values of 2na 

 and a'~ — d^, by rejecting the common factor A C, (3) is reduced 

 to a' + d — xa'^Lx'd=0, or a' {I — x) + d {\ — a;') = 0, or since 

 a' =b + d'^^A C+CB' (using the upper sign when B' is not be- 

 tween A and C, and — when it is between A and C), we get 

 AC{l—x)±CB'{l—x-\-l—x)=0, (4). Now it is evident 

 that C B' must be arbitrary, and not dependent on ^ C or 1 — a:, 

 1 — a;'; .-, we must have 1 — x -}- 1 — a?' = 0, and (4) is reduced to 

 A C {1 — a;) = 0, which, since ^ C is not = 0, gives 1 — a; = 0, 

 .-. 1 — x' = 0, or a;=l, a;'=l; hence (1) and (2) become c- = 

 a^ + a'^ — 2naa', (1'), c^ = a^ ■}- d^±2na d, (2'). In like manner, 

 by regarding the angle A as belonging to the isosceles triangle 

 ABC, we get from the triangles ^ B ^', B C'B\ c^ = a^ + a'~ — 

 2n'aa\ (1"), c^ = a- + d'^ ±2n'a'd\ (2"); where for + we must 

 use — when B is between A and C, and -\- when B is in -4 C" pro- 

 duced beyond C". By equating the values of c^, as given by (1') and 

 (1"), we get n =^n, or -r^ = -t-^i as was to be proved. It is evi- 

 dent that ABE may represent any right triangle having A for one 

 of its acute angles, and its hypothenuse on A G ; also A B'E' may 

 denote any right triangle which has A for one of its acute angles, and 

 its hypothenuse on AF; hence, from what has been shown, n will be 

 the same for all the triangles represented by A B E and A B'E' ; 

 that is, all right triangles which have a common or equal acute angle 

 will have equal values of n corresponding to the common or equal 

 acute angle. There is one case that apparently forms an exception 

 to what has been shown ; and that is when the hypothenuse of a tri- 



