CHAP. V] RATIONAL TRIANGLES. 193 



L. Euler 7 noted that in any triangle with rational sides a, b, c, and 



rational area, 



_ (ps qr)(pr T gs) . p 2 + g 2 . r 2 + s 2 



pgrs pq rs 



and that every pair of sides are in the ratio of two numbers of the form 

 (a 2 + 2 )/aft since 



r 2 _1_ o2 ~2 _}_ 7,2 



a : b = : - , if x = ps db qr, y = pr =F gs, 

 rs 



whence 



The portion of Euler's paper containing his derivation of (1) is missing. 

 It is probable that he employed Bachet's method of juxtaposing two right 

 triangles, using those with the sides 



2 __ z i _ 2 2 2 2 - 2 







pq pq rs rs 



and obtaining (1) with the upper or lower signs according as the component 

 triangles do not or do overlap. 



J. Cunliffe 8 juxtaposed two right triangles with a common side 2rs = 2mn 

 and hypotenuses r 2 + s 2 , w 2 + n 2 . 



J. Davey 9 found three triangles with integral sides and areas having 

 equal perimeters and areas in the ratio of a = 2, b = 7, c = 15. Let the 

 triangles be AFB, BFC, CFD with collinear bases and the common altitude 

 FE. Take 



Then AE = (r 2 - l>/(2r), etc. By the equality of the perimeters, 



s 2 - 1 1 t 2 - 1 1 



-=r-, - = s -- . 

 s t t u 



Then the conditions that the bases be proportional to a, b, c reduce to 

 (ar 2 + 6)/r = (at 2 + V)/t, whence r = b/(a) (since r =M)> and to u = c/(bs). 

 Eliminating r, u, s between our four relations in r, u, s, t, we get 



* 4 - (d 2 + de + 2)t 2 + de + l = 0, d = C - , e=- -. 



c a 



For a = 2, 6 = 7, c = 15, we get the rational root t = 5/3. Taking 

 v = 420, we have AF = 541, BF = 525, CF = 476, DF = 421, AB = 26, 

 BC = 91, CD = 195, perim. = 1092. 



To find a triangle ABC with integral sides and area such that the dis- 

 tances from A, B, C to the center of the inscribed circle shall be integers, 



7 Comm. Arith. Coll., II, 1849, 648, posthumous fragment. Same in Opera postuma, 1, 



1862, 101. 



8 The Gentleman's Math. Companion, London, 3, No. 15, 1812, 398. 



9 Ladies' Diary, 1821, 36-7, Quest. 1364. 

 14 



