208 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v 



The area being rational, we may set (Schubert 47 ) 



77 (j ?y 



tan \a = , tan |/3 = - , tan |0 = - , 

 m p x 



where m, n are relatively prime integers, etc. Hence 



x~ y* m? ri* p 2 q 2 

 2xy 2mn 2pq 



2(x 2 y^mnpq = xy(mp + nq)(mq np). 

 It is concluded erroneously 90 that the only integral solutions are 



(x, y) = (mq, np) or (mp, nq). 



Hence there remains in doubt his conclusion that no Heron triangle has 

 more than one rational median. 



R. Giintsche 91 considered a triangle ABC whose sides a, 6, c, area I and 

 median CF are rational. If s is the semi-perimeter and p the radius of the 

 inscribed circle, 



cot %A = s(s a)//, sp I, 



so that the cotangents a, 13, 7 of %A, %B, %C must be rational. Also, 

 a + ft + 7 = apy. Taking p = (a/3 1 )/(/?), we have 



s = a + /3, a = (3 +-, 6 = a + - , c = s - - ' - - = 7. 



p a a p 



Let F be the center of AB and v = cot %(CFB). From triangles CAF and 

 CFB we obtain the two values of c/2 : 



(3) a _l + l_^ = ,_l + ^_l. 



a v v p 



To secure symmetry, set /?' = 1//5. We obtain 



(4) 2v*p-'a - - v(p-'*a + /3'a 2 --/?'- a) - 2/3'a = 0, 



which is quadratic in each of v, a, /?'. Taking a as a parameter, we may 

 treat the equation in v, @' by Euler's 144 method of Ch. XXII. But the 

 second value of v belonging to /3' is 1/v, so that the corresponding angle 

 has been increased by TT. To obtain an essentially new solution, introduce 

 the variable = v$' in place of v before applying Euler's process. A 

 similar remark holds for the more general equation 



(5) px 2 y + qxy z + rxy + hqx + hpy = 0. 



90 Other sets of solutions are m = 2, n = 1, p = 2, q = 1, x = 2, y = 1 or x = 1, y = 2; 



m = 2, n = 1, p = 3, q = 1, x = 3, y = 4 or x = 4, y = 3. For x = mq, y = np, 

 the factor mq np may be cancelled from the equation in the text, giving p(m 2n) 

 = q(2m n). Hence 2m n = Ip, m 2n = Iq, where I = 1 or 3 (m and n being 

 relatively prime). Schubert erroneously excluded I = 3, an example for which is p = 3, 

 q = 1, m = 5, n = 1; this however does not affect the relation between tan (a/2) and 

 tan (/3/2). 



91 Sitzungsber. Berlin Math. Gesell., 4, 1905, 27-38. 



