134 Mb WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULAE, 



Let ji, q, r be given numbers which express the given ratios of 

 X, y, X to one another, viz., 



X : y =1) : q, x : % =p : r, y : z =q : r, ' 



then a' : h' =pa : qh, a' : c' =pa : re, V : c'^qb : re. 



Since we may give one of the lines «', b', c' any magnitude, let us 

 assume that 



a'=pa; then b' = qb and c' = re. 



Thus a, b', e' the sides of a triangle are given, from which the oppo- 

 site angles A', S', C may be found, and therefore are known ; and 

 since A, B, C are known, a = A + A', fi = B+B', y = C+C' are known; 

 the problem is now made identical with that of Art. 7. and the for- 

 mulse which apply to the one, resolve also the other. 



Since the lines «', b', c are the sides of a triangle, any two of them 

 must be greater than the third ; and unless this condition be satisfied, 

 the problem will not admit of a solution in that particular case. 



The angles of a triangle, whose sides a, b, c are given, may be 

 most conveniently found from this formula. 



Fmd s = ^{a + b + c), R=\/r- ' ^ ^ ' ^ '\; 



Tt R R 



then tan hA = : tan 4 JS = — r ; tan i C = . 



^ s—a ^ s—b ^ s—c 



This way of finding the angles gives the advantage of a verification, 

 viz. ^{A-\-B + C) = 90°. The line R is the radius of a circle inscribed 

 in the triangle*. 



33. Since the function R may be either positive or negative, the 

 angles A', B', C may be either all positive or all negative-, hence 

 a, /3, 7 will have each two values, viz., 



a, = A + A', li.^B + B', 7, = C+C', 



a, = A-A', (i, = B-B, y,= C-C'. 



• There is a formula perfectly similar to this for spherical triangles. 



