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



Use may be made of the expansion of cos kB in terms of cos B or of 



2 cos kB = (x + z 2 - 1)* + (x - x*- )*, x = cos B. 

 K. Schwering 112 took any linear relation between the angles. 



MISCELLANEOUS RESULTS ON TRIANGLES WHOSE AREA NEED NOT BE 



RATIONAL. 



A. Girard 112a noted that z=B 2 +BD+D 2 , x = 2BD+D 2 , y = 2BD+B 2 



are sides of a triangle in which an angle is 60 [i.e., satisfy z 2 = x 2 xy+y 2 ], 

 and the same is true of z, Xi B 2 D 2 , y. Also z, x, Xi are sides of a triangle 

 in which an angle is 120 [i.e., z 2 = x 2 + xx\ + a;?]. 



To find integral sides a, b, c of a triangle ABC such that, if P is the point 

 within it from which the sides subtend equal angles, the distances x = AP, 

 y = BP } z CP are expressed by integers, we have 



c 2 = x 2 -f- xy + y 2 , b 2 = x 2 + xz + z 2 , a 2 = ?/ 2 + 2/2 + z 2 . 



Many solvers 113 took c = x + y m, b = x + z n and obtained two 

 values for x, from which we get z = (hy mri)/(y /c), where h and k are 

 known. Then 



(7? 2fr \ 2 



2/ 2 + 2~- 2/ + W71J 



determines $/ rationally. Cf. papers 116 and 123; also 65, 67, 68, 70-73 

 of Ch. XIX. 



Berton stated and J. de Virieu 114 proved that the area of a triangle is 

 not rational if the sum of the sides, without a common factor 2, is odd. 



W. S. B. Woolhouse 115 proved that, if three numbers ^ n are taken at 

 random from a list of such triples and if p n is the probability they will be 

 sides of a possible triangle, then p n , p n +i, p n +z are in arithmetical progression 

 if n is even. He found the probability that three integers ^ n named by 

 three different persons or by the same person will be proportional to the 

 sides of a real triangle. 



S. Bills 116 found the least integral sides BC, CA, AB of a triangle for 

 which x = OA, y = OB and z = OC make equal angles and are measured 

 by integers. 113 First, AB 2 = x 2 + xy + y 2 = D, AC 2 = x 2 + xz + z 2 = D if 



p 2 - 1 q* 1 



Take q = 2. Then BC 2 = y 2 + yz + z 2 = D if 25p 4 + = D, which 

 holds if p = 9/4, whence x = 440, y = 325, z = 264. 



H. S. Monck 117 gave a very special discussion of the problem to find 

 the least triangle with sides in arithmetical progression and altitudes in 



112 Archiv Math. Phys., (3), 21, 1913, 129-136. 



1120 L'Arith. de S. Stevin ... par A. Girard, Leide, 1625, 676; Les Oeuvres Math, de S. 

 Stevin, par A. Girard, 1634, 169. 



113 The Lady's and Gentleman's Diary, London, 1844, 50-1, Quest. 1705. 



114 Nouv. Ann. Math., (2), 3, 1864, 168-170. 



116 Math. Quest. Educ. Times, 9, 1868, 63-5, 91-2. 

 Ibid., 20, 1874,60-1. 



117 Ibid., 21, 1874, 108-9. 





