CHAP. V] TRIANGLES WITH RATIONAL MEDIANS. 203 



We have g 2 - h 2 = 3(c 2 - 6 2 ). Euler took 



g + h = 3pq, g h = rs, c + b = pr, c b = qs. 



From 



g 2 + h 2 = 4a 2 + b 2 + c 2 , / 2 = 2c 2 + 26 2 - a 2 , 



we get, on setting p = x -\- y, s = x y, 





2Mxy, = x 2 



5g 2 - r 2 5r 2 - 



^" - ~ ^ V - 



A > 



4r 2 

 Take afq = x + ty, f/r = x + uy. Then 



i - ^ 2 



y '' 2(t- M) ' 2(u-- N)' 

 All conditions are satisfied if we take 



N - M x _ (M - AQ 2 - 4 



2 ' 



The cases r = q and r = 3<? are excluded since M + TV" =4= 0. For 

 5 = 1, r = 2, we obtain the solution given above. For q = 2, r = 1, we get 



a = 404, 6 = 377, c = 619, / = 3-314, g = 3-325, h = 3-159. 



Euler's 70 paper of 1779 does not differ materially frbm the preceding. 



Euler's 71 paper of 1782 avoided the earlier restrictions on the generality 

 of the solution. Changing the notations to conform with his earlier ones, 

 we may set 



h + g = ~(b-c), h-g=^-(b + c). 

 p a 



From 



(h + g} 2 +(h- g) 2 = 2h 2 + 2g 2 = Sa 2 + (b + c} 2 + (b - c) 2 , 

 we get 



Then / 2 = (b + c) 2 + (b c) 2 a 2 gives a similar formula for / 2 . Write 



Then 



(2) ^ = 7 2 + 5 2 + 2P T 5, j = 7 2 + 5 2 



Take 7 = 4(P + Q), 5 = (P - Q) 2 - 4. Then (2) are the squares of 

 (P - Q)(3P + Q) - 4, (Q - P)(3Q + P) - 4. 



70 Mem. Acad. Petrop., 2, 1807-8, 10; Comm. Arith., II, 362-5. 



71 Mem. Acad. Petrop., 7, 1820, 3; Comm. Arith., II, 488-91. 



