CHAP. V] TRIANGLES WITH A RATIONAL MEDIAN. 207 



Subtract, replace sx by (s d)x + (s fyx + (s c)x, and c b by 

 s b (s c). Thus 



s as b s c 

 x x + 1 x 1 



Since s a, etc. shall be positive 1 < x < 1. Similarly, the rationality 

 of t b implies the existence of a rational value y, I < y < 1, for which 



s b s c s a 

 y y + 1 y - 1 



The two equations determine the ratios of s a, . We may set 



s _ a = ( x + 2y + l}x(l - y) = A, s - b = (2x + y - 1)(1 + x)y = B, 

 s - c = (x - y + 1)(1 - a;)(l + y) = C. 



By addition, s = 3xy + x y + 1- Hence for any proper fractions x } y, 

 we have rational values of a, b, c, and of 



2t a = sx + (s a)fx, 2t b = sy + (s b}/y. 



For x = |, y = i we find a = 17, b = 27, c = 16, 2t a = 41, 2t b = 19. 



If also t c is to be rational, we must have a rational solution z, 1 < z < 1, 

 of 



s c s a s b 



, ; H T U. 



2 2+1 21 



Replacing s a, s b, s c by their values A, B, C, we obtain a rela- 

 tion R between x, y, z, quadratic in each. Now the pair of equations 



- y} _ B = (x-y + l)x(l - y) C 



1+2 "l-2~ 1+2 2 



have the sum R and are such that the elimination of 2 gives 



y = (7 - 4x - 2z 2 )/(l(te - 5). 



He gave eight further pairs of equations with the sum R such that the 

 elimination of 2 yields an equation linear in x or y. For the problem of 

 three rational medians, this method lacks the generality and simplicity 

 of Euler's. 71 



HERON TRIANGLES WITH A RATIONAL MEDIAN; HERON PARALLELOGRAMS. 



H. Schubert 89 defined a Heron parallelogram to be one whose sides, 

 diagonals and area are rational. Call a, /3 the angles made by a diagonal 

 with the concurring sides a, 6; and 6 the angle between the diagonals and 

 opposite b. Then 



a : b = sin (0 + 0) : sin (6 a), a sin a = b sin 0, 



the second following from the equal areas on each side of our diagonal. 

 Hence 



2 cot 9 = cot a cot ]8. 



89 Auslese Unterrichts- u. Vorlesungspraxis, Leipzig, 2, 1905, 36-45. Unterrichtsblatter 

 Math. u. Naturw., 6, 1900, 70-1. Schubert, 47 21-26. 



