210 



HISTORY OF THE THEORY OF NUMBERS. 



[CHAP. V 



N. Fuss 95 investigated triangles with rational sides a, b, c, rational 

 angle-bisectors a, /?, 7 and rational area cr. The altitudes are then rational. 

 Set 



b + c a = 2f, a + c b = 2g, a -\- b c = 2h. 

 Then 



a + b + c = 2(f + g + h), a 2 = (/ + g + h)fgh. 



He took / = pq, g = qr, h = pr. Then a is rational if 



pq -}- pr -}- qr = s 2 , 

 where s is rational. Since a = g + h, b = f + h, c = f + g, we get 



A/6c(6 + c + a) (b + c a) 2pqs r. r-. -r 

 b -f- c pQ' -f- s 2 



The quantity under the last radical equals r 2 '+ s 2 , which is therefore to be 

 a square. Similarly, p 2 + s 2 and q 2 + s 2 are to be squares. Set p = Is, 

 q = ms, r = ns. Then 1 + I 2 , etc., are to be squares, while 



Im + In + WTO = 1. 

 These conditions are satisfied if 



& " 



P 2 - 



1 Im 



n = 



2PQ 2RS l^-m 



For example, let P = R = 2, Q = S = 1. Then l = m = 3/4, n = 7/24. 

 Take s = 1 and multiply a, a, etc. by 32. We get 



a = 14, b = c = 25, a = 24, = 7 = " Rrt 

 J. Cunliffe 9 * noted that the triangle with the sides 



mn(m 2 n 2 )(r 2 + s 2 ) 2 , rs(r 2 s 2 )(w 2 



f /vy)/yi [ /y*2 _ . o2 ^ . _ <yo ( /VViS . _ /yj2 j ) ( ( /y*2 _ . o2 ^ ( /yyj2 . _ /vj 

 | llll(j\l O J I O^//6 * * / ) I V ** / V'* * 



has rational area and angle-bisectors. He 97 obtained such a triangle with 

 the sides 39, 150, 175 by taking three right triangles (m 2 + n 2 , m 2 n z , 



143 



32 



2mn) with a common leg 2mn, where m, n = 12, 1; 6, 2; 4, 3 and using 

 each right triangle twice. 



96 M6m. Acad. Sc. St. PetSrsbourg, 4, 1813, 240-7. 



96 New Series of the Math. Repository, London, 3, Pt. 2, 1814, 13-15. 



97 Ibid., 4, Pt. 2, 1819, 64. 



