CHAP. V] TRIANGLES WITH RATIONAL MEDIANS. 205 



Equate it to (2x + y m) 2 and the similar expression for 4BE 2 to 

 (x + 2y n) 2 . Solve the two resulting linear equations for x, y in terms 

 of d, m, n. Reject the common denominator. Thus 



x = d 2 (4n 2m) + 2d(m 2 n 2 ) mn(2m n), 

 y = d 2 (4m 2n) 2d(m 2 n") -f- mn(m 2n), 

 z = 2d 2 (m + n) Qmnd + mn(m + n). 



Then 4CD 2 = 2(x 2 + 2/ 2 ) 2 2 becomes a quartic in d which is a square for 



3(m + ri)(m - n) 2 (2m 2 - - 5mn + 2n 2 ) 

 10(m - nY - mn(m 2 + n 2 ) 



C. Gill 78 gave a solution in which the sides are proportional to expres- 

 sions in the sines and cosines of two of the angles A, B, subject to the condi- 

 tion that tan A/2 equals one of four complicated functions of sin B and 

 cos B. The numerical example is the same as the first one of Euler's 68 

 paper of 1773. 



E. W. Grebe 79 thought the problem was a new one. Changing his nota- 

 tions to conform with Euler's, we see that 2b 2 + 2c 2 a 2 = / 2 implies 



From this and a similar formula involving g, we get 



b + c + / = m(a + b - c), 6 + c - / = : - (a - b + c), 



iHt 



c + a + g = p(b + c - a), c + a - # = - (6 - c + a), 



where m and p are unknowns. These four relations determine the ratios 

 of a, b, c, /, g as rational functions of m and p. Then 2a 2 + 26 2 c 2 (which 

 is to equal /i 2 ) is made a rational square by choice of p rationally in terms 

 of m. Then the sides and medians are quintic functions of m. 



C. L. A. Kunze 80 gave essentially the solution in Euler's 71 paper of 1782. 



J. W. Tesch 81 gave Cunliffe's 75 solution. 



* E. Haentzschel 82 and Schubert 88 treated the problem. Cf. papers 101, 

 106. 



The medians of a triangle with rational sides a, b, c are proportional to 

 the sides if and only if a 2 -f c 2 = 26 2 ; such a triangle is called automedian. 

 Reports of many papers on this equation are given in Ch. XIV. 



TRIANGLES WITH A RATIONAL MEDIAN AND RATIONAL SIDES; PARALLELO- 

 GRAMS WITH RATIONAL SIDES AND DIAGONALS. 



C. G. Bachet's 3 fourth problem, added to his comment on Diophantus, 

 VI, 18, was to find a rational triangle with one rational median. First 



78 Application of the angular analysis to the solution of indeterminate problems of the second 



degree, New York, 1848, 50-2. Results quoted in 1'intermediaire des math., 5, 1898, 10. 

 Cf. A. Martin, Math. Quest. Educ. Times, 25, 1876, 96-7; E. Turriere, 1'enseignement 

 math., 19, 1917, 267-272. 



79 Archiv Math. Phys., 17, 1851, 463-74. 



80 Ueber einige Aufgaben aus der Dioph. Analysis, Progr. Weimar, 1862, 9. 



81 L'intermediaire des math., 3, 1896, 237. Repeated, 20, 1913, 219. 



82 Jahresber. d. Deutschen Math.-Vereinigung, 25, 1916, 333-351. 



