202 HISTORY OF THE THEORY OP NUMBERS. [CHAP, v 



He found a pentagon inscribed in a circle with rational sides and areas for 

 all the triangles into which the pentagon can be divided by diagonals. 



TRIANGLES ALL OF WHOSE SIDES AND MEDIANS ARE RATIONAL. 

 L. Euler 68 denoted the sides by 2a, 2b, 2c, and the medians by /, g, h. 



Then 



2& 2 + 2c 2 - a 2 = f 2 , etc., 2g 2 + 2h 2 - f 2 = 9a 2 , etc. 



Hence, if 2/, 2g, 2h be taken as sides of a triangle, its medians are 3a, 3fr, 3c. 

 Write a- = a + b + c. Then 



(b - c) 2 + er(6 + c - a) = / 2 , (a - c) 2 + a(a + c - 6) = 2 . 

 Set / = 6 c + (rp, g = a c + aq. Then 



6 + c - a = 2(6 - c)p + vp 2 , a + c - b = 2(a - c)q + aq 2 . 



Solving each for c and adding a + b, we have two expressions for or. Equat- 

 ing these, we get the ratio a' : b' of a : b. Euler took a! = a and got 



a = 1 + q p 2 2pq p 2 q + 2pq 2 , b = 1 + p q 2 2pq pq 2 + 2p 2 q. 



Then <r/2 = 1 + p + q 3pq, so that c is known and hence also /, g. 

 Next, 



h*= (a- b) 2 + a(a + b - c) = Ay + 2g 3 + Cg 2 + 2Dg + E 2 , 

 where 



A = 1 + 3p, 5 = - 1 + Up - 9p 2 - 9p 3 , 



C = - 3(1 + 2p - 2p 2 + 6p 3 - 3p 4 ), D = 2 - 9p - 3p 2 + lip 3 + 3p 4 , 



^ = 2 + p - p 2 . 



We can obtain rational solutions by setting 



B D 



+-rqE or Aq 2 qE. 

 A & 



Euler examined the simplest cases p = 2(p = 0orl being excluded). 

 For p = - 2, we have A = - 5, B = 13, C = 321, D = - 32, E = - 4. 

 Taking the second expression for h, we have 



ft = - 5g 2 - 8? + 4, g = ^ , /i = *}* 

 Multiplying the resulting values oi a,b, - by 4/3, we get 

 a = 158, 6 = 127, c = 131, / = 204, g = 261, h = 255. 



Since f /, f gr, f /& are sides of a triangle with the medians a, 6, c as remarked 

 at the outset, we get the new solution 

 a = 68, 6 = 87, c = 85, / = 158, g = 127, h = 131. 



Euler's 69 paper of 1778 deals with triangles in which the distances of the 

 vertices from the center of gravity are rational and the sides are rational. 



68 Novi Comm. Acad. Petrop., 18, 1773, 171; Comm. Arith. Coll., 1, 1849, 507-15. 

 89 Nova Acta Acad. Petrop., 12, 1794, 101; Comm. Arith., II, 294-301. 



