CHAP. V] TRIANGLES WITH RATIONAL ANGLE-BISECTORS. 213 



H. Schubert 47 (pp. 17-21, or Schubert, 88 27-36) considered a Heron 

 triangle ABC in which the bisector w a of angle A is rational. Since it 

 divides the triangle into two Heron triangles we need only take A/2 and B 

 to be Heron angles, i. e., 



A 2uv A u 2 - v 2 2pq p* - g 2 



sin = -T ; , cos = -- , sin B = - , cos B = - . 

 2 u 2 + v 2 2 v? + v 2 p z + q 2 p 2 + q 2 



Thus sin A and cos A are rational, so that in his 47 formulas for the sides of a 

 Heron triangle we need only take m = u 2 v 2 , n = 2iw. To make w a 

 and w 6 (and hence w e ) rational, take both A/2 and B/2 as Heron angles. He 

 considered (6) Heron triangles with both a rational bisector and a rational 

 median. 



An anonymous writer 107 gave three large integers which are the sides 

 of a triangle having integral values for the area, three interior and three 

 exterior angle-bisectors and the 12 segments cut off by them on the opposite 

 sides. Also a triangle having integral values for the sides, area, altitude 

 and two bisectors from the vertex, and the four segments of the base cut 

 off by the two bisectors. M. Rignaux 108 gave a solution hi smaller integers 

 of the last problem. 



E. Turriere 109 considered a triangle with rational values for the sides 

 a, b, c and bisector d of the interior angle A. Thus 



6+c b+c 1 



y 2 = nx 2 + 1. y = - , x = --- -a, n = . 



a a oc 



The rational solutions of this Pell equation are 



t 2 + n 2t 



- X ~ 



t 2 -n' 



Hence the desired triangle is obtained by assigning any rational values to 

 b, c and taking a = (be t 2 )(b + c)/q, d = 2bct/q, q = be + t 2 . In a Heron 

 triangle, the bisector of angle A is rational if and only if tan \A is rational. 

 Every Heron triangle whose bisectors are rational is the pedal triangle to 

 a Heron triangle. 



* O. Schulz 157 (pp. 72-3) treated rational triangles with three rational 

 angle-bisectors. 



TRIANGLES WITH RATIONAL SIDES AND A LINEAR RELATION BETWEEN 



THE ANGLES. 



K. Schwering 110 discussed triangles with integral sides one of whose 

 angles is double another. 



J. Heinrichs 111 generalized the problem, taking the relation a = n[3 + y 

 between the angles. Set B = (3/2. Then 



a : c : b = cos (n 1}B : cos (n + 1) B : 2 cos B Vl cos 2 B. 



107 L'interm<diaire des math., 23, 1916, 51-2, 73. 



108 Ibid., 234-7. 



109 L'enseignement math., 18, 1916, 397-407. 



110 Gymn. Progr., Coesfeld, 1886. 



111 Zeitschr. Math. Naturwiss. Unterricht, 42, 1911, 148-153. 



