CHAP. V] TRIANGLES WITH RATIONAL ANGLE-BISECTORS. 209 



which includes the case treated by Kummer. 133 To simplify Euler's process, 



set 



Y -- Y -- 



* ' '"' 



From the initial pah- x = x , y = yo, we form 



Then x,, y : is a new pair of solutions of (5). Similarly, we may start 



with x , YQ. For (4), 



h = _ 1, p = 2a, q = - a, r = 1 - a 2 , 0() = - ( + 2)/(2$ + 1). 



Hence from the initial pair v 0) fi' , we get PI = fi'oQ(vA}> P'I = Po^Wo). 

 From the trivial solution a = p, v = 1, 0i = 1/p, we get 



- (2p + 1) , 



From these we obtain a new set; etc. We may replace v by 1/v, since 

 (3) remains unaltered; we obtain the solution 



" 







6 = (p + 2)(2p + 1)(P 2 + 1), c = 2(p 2 - l)(p 2 + p + 1), 



*(p + 2) 2 + (2p + I) 2 , / = ^p(p 2 - l)(p + 2)(2p + l)(p 2 + p + 1). 



E. Haentzschel 92 repeated Guntsche's deduction of (3), with a, inter- 

 changed. For symmetry replace the new a. by its reciprocal. Hence 



a z - 1 ff 2 - 1 r 2 - 1 



2a 

 The value obtained by solving for v will be rational if 



2 = D. 



This quartic in is treated by use of Weierstrass's elliptic ^-function 

 [cf. Haentzschel 82 of Ch. XV]. There result various particular types of 

 Heron parallelograms. 



TRIANGLES WITH RATIONAL SIDES AND ONE OR MORE RATIONAL ANGLE- 



BISECTORS. 



C. G. Bachet 93 gave a long construction and discussion leading to the 

 special acute angled triangle with the sides (reduced 1 : 4) 20, 20, 5 and 

 having 6 as the bisector of either equal angle; also the oblique angled tri- 

 angle with the sides 80, 125, 164 and having 60 as the angle-bisector drawn 

 to the side 164. [The area of each triangle is irrational.;] 



J. Kersey 94 discussed oblique triangles with rational sides and area and 

 one rational angle-bisector or median. 



92 Sitzungsber. Berlin Math. Gesell., 13, 1913-4, 80-9. 



93 Diophanti Alex. Arith. 3 . . . , 1621, 419-21. Ed. by S. Fermat, 1670, 317-9. 



94 The Elements of Algebra, London, Books 3 and 4, 1674, 144-8. 

 15 



