. Carnegie Institution of Washington publication. CHAP. V] RATIONAL QUADRILATERALS. 217 From one inscriptible quadrilateral we get two others (but not with per- pendicular diagonals) by permuting the sides. The area of each of the three quadrilaterals is the product of the three distinct diagonals divided by double the area of the circumscribed circle (A. Girard; proof by Grebe, Manuel de Gm., 1831, 435). L. N. M. Carnot132 noted that the segments of the diagonals of a quad- rilateral are expressible rationally in terms of the sides and diagonals. E. E. Kummer133 noted that Chasles unriddled t
. Carnegie Institution of Washington publication. CHAP. V] RATIONAL QUADRILATERALS. 217 From one inscriptible quadrilateral we get two others (but not with per- pendicular diagonals) by permuting the sides. The area of each of the three quadrilaterals is the product of the three distinct diagonals divided by double the area of the circumscribed circle (A. Girard; proof by Grebe, Manuel de Gm., 1831, 435). L. N. M. Carnot132 noted that the segments of the diagonals of a quad- rilateral are expressible rationally in terms of the sides and diagonals. E. E. Kummer133 noted that Chasles unriddled the obscurity of Brah- megupta without perceiving the method used by the latter, and expressed Brahmegupta's theorem in the following form. If the four sides of a quadrilateral, inscriptible in a circle, have the values. (a2 + 62)(c2 - fore sin u/sm v. But a _ sin w d sin w ft a sin u /3 sinu' d sin v ' d d sin v' Hence /3/5, 1 + /3/5 = BD/d, 8 and /3 are rational. Similarly, a and 7 are rational. Next, c = cos w is rational, in view of (1) a2 = a2 + /32 - 2a(3c. Set c = mjn, where m, n are relatively prime. Without loss of generality, we may assume that a, a, (3 are integers with no common factor. To treat one of two analogous cases leading to like results, let n be odd. Then n must divide a/3. Thus a = rai, 0 = sft, n = rs, (2) a2 = r2al + s2/3? - 2ma1j3l. Now «i, @i are relatively prime, since a common factor would divide a. We may take ft odd. The product of (2) by r2 may be given the form ^1^2 = (n2 - m2)/3?, Fi = ar + r2^ - wft, F2 = ar - r2«i + raft. If FI and F2 were both divisible by a prime factor p of ft, then 2r2ai and hence ra\ would be divisible by p, likewise a by (2), whereas a, a, /3 do not 182 G6om6trie de position, Paris, 1803, 391-3. 133 Jour, fur Math., 37, 1848, Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfect
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