CHAP. V] RATIONAL TRIANGLES. 197 



D. S. Hart 23 juxtaposed two right triangles with the common leg 2pr 

 and further legs r(p* 1), p(r 2 1), and obtained 



(p + r)(pr-l), r(P 2 + l), P(r 2 + 1), 



viz., (1) for the case of the upper signs and q = s = 1. The last assumption 

 does not restrict the generality of the result. 



Hart 24 noted that the triangle with the sides w 1, w, w + 1 has a 

 rational area if 3w 2 12 = D. He obtained w = nfd, d = x 2 3y 2 and 

 took d = 1, whose general set of solutions is known. 



A. B. Evans 25 found a triangle whose sides a, 6, c, radii 



x = fr(l + tan J^)(l + tan )/(! + tan |C), 



y, z of Malfatti's circles, and radius r of the inscribed circle are all rational. 

 Take cot %A = m/n, cot |5 = p/q, m 2 + n 2 = D, p 2 + q 2 = D. Then 

 tan \A, etc., are rational. A. Martin took cot \C = 3, cot \B = 4; then 

 the ratios of x } y, z, a, b, c to r are known. 



H. S. Monck 26 showed how to deduce a second from one triangle with 

 integral sides, two differing by unity. 



J. L. McKenzie 27 found a triangle whose area and sides are integers, 

 semi-perimeter is a square, two sides having a given common difference. 



D. S. Hart 28 discussed rational triangles two of whose sides differ by 

 unity. 



R. Hoppe 29 discussed triangles with the sides n r, n, n + r and 

 rational area A. Thus A = jww, where 3m 2 = n 2 4r 2 . Hence n is even, 

 n = 2p, and m = 2q, whence p 2 3g 2 = r 2 . First, let r = 1. If p k) qk is 

 a solution in integers, then is also 



p k+l = 2p k + 3q k , q k +i = p k + 2q k . 

 Further, p k+ i 4p k + p k -i = and similarly for the g's. Hence 



s k (2 + 3)* = s c . 

 The resulting values of n, A are 



^O 



n = (2 + VS)* + (2 - V3)*, A = - {(2 + V3) 2 * - (2 - A/3) 2 *}, 



for k = 0, 1, . It is proved that there are no further solutions. 



Next, let r be undetermined. Then p : r = 3X 2 + /* 2 : 3X 2 /* 2 , where 

 X and n are relatively prime integers. Thus the sides are 



_ 3(X 2 + M 2 ), 2(3X 2 + M 2 ), 9X 2 + M 2 . _ 



23 Math. Quest. Educ. Times, 23, 1875, 108. 



24 Ibid., 23, 1875, 83-4. 

 26 Ibid., 22, 1875, 70-1. 



26 Ibid., 24, 1876, 36-8. 



27 Ibid., 25, 1876, 105-6. 



28 Ibid., 28, 1878, 66-7. 



29 Archiv Math. Phys., 64, 1879, 441. 



