CHAP. V] RATIONAL QUADRILATERALS. 219 



Determine all rational solutions 134 x, z of (7). Then (5) determines two 

 rational values of y, and (3), (4), (6) give the segments of the diagonals and 

 the sides as rational numbers. 



W. Ligowski 135 and J. Cunliffe 135a gave special rational inscribed quadri- 

 laterals. 



D. S. Hart 136 desired an inscriptible quadrilateral with integral sides 

 a, b, c, d and diagonals x, y. Thus xy ac + bd, x : y = be + ad : ab -f- cd, 

 so that the product of the three sums is to be a square, say the square of 

 abc + d(a? + fe 2 + c 2 )/2, which determines d. A. B. Evans took the sides 

 AB = x, BC = mx, CD = nx, AD = px. As known, 



AC 2 = (mp + rc)az 2 , BD* = (mp + ri)a*/a, a = + mn 



m + pn 

 The last gives p rationally. Let 



a = a 2 , n = q 2 , mp -\- n = [q + my/(a?q 2 I)} 2 , 



which gives m. Hart 137 found a trapezoid with integral values for the sides, 

 diagonals, area and perpendicular between the parallel sides. 



G. Darboux 138 based a geometrical theory of quadrilaterals upon two 

 equations 



0*1 + bt 2 + ct 3 + dU = 0, ? + 7 + T + 7 = > 



l\ &2 's *4 



where a, b, c, d are the sides, and ij = e l % co/ being the angle between 

 the jth side and any line in the plane. Regarding the 's as homogeneous 

 coordinates, we have a plane cubic curve. 



O. Schlomilch, 139 started with two right triangles T a =(l-a*, 2a, 1+a 2 ) 

 and Tp, reduced their sides proportionally to obtain a common leg, 

 and juxtaposed them to obtain a triangle with the sides (1 + a 2 )/3, 

 (a + j8)(l a/3), (1 -f /3 2 )a. Treating two such oblique triangles similarly, 

 we obtain a quadrilateral with the sides (1 + cr)/3, (1 + /3 2 )a, (1 + 7 2 )5e, 

 (1 + 6 2 )7e, where 



(y 4 



The sides, diagonals and area are rational if a, , 5 are. 



S. Robins 140 listed rational trapeziums whose area equals the square 

 root of the product of the four sides, found by use of convergents to Va 2 + 1. 



H. Schubert 88 (pp. 49-54) considered quadrilaterals inscribed in a circle 

 of radius r. Let 2i, -, 2a 4 be the arcs subtended by the sides. Then 



134 From simple solutions of (7), Kummer obtained new solutions by the method of Euler 143 " 145 



of Ch. XXII and thus deduced various rules for forming rational quadrilaterals. 



135 Archiv Math. Phys., 47, 1867, 113-6. 



1350 New Series of Math. Repository (ed., T. Leybourn), 2, 1809, I, 74-5, 225-6. 



136 Math. Quest. Educ. Times, 20, 1874, 64-5. 



137 Ibid., 80-81. For history of inscriptible quadrilaterals with given sides, 21, 1874, 29-35. 



138 Bull. Sc. Math. Astr., (2), 3, 1, 1879, 109-128; Comptes Rendus Paris, 88, 1879, 1183, 1252. 



139 Zeitschr. Math. Naturw. Unterricht, 24, 1893, 401-9. 



140 Amer. Math. Monthly, 5, 1898, 181-2. 



