(If you find that the solutions are too numerous, choose the solutions which provide the smallest perimeter and area.)

Source: Original.

Solutions were received from Joseph DeVincentis, Luc Kimp, Al Zimmermann, Denis Borris, Philippe Fondanaiche, Sandy Thompson, and William Thome. Denis Borris' and Al Zimmermann's solutions follow:

Sides of triangle = a b c Sides of rectangle = x y Let j = (a+b+c)/2 (1) Let k = sqrt[j(j-a)(j-b)(j-c)] (2) Since a+b+c = 2(x+y), then j=x+y (3) Since xy = (2), then xy = k (4) Combining (3) and (4): j - y = k / y ; y^2 - jy + k = 0 Then: y = [j +- sqrt(j^2 - 4k)] / 2 (quadratic) This equation permits a reasonable a b c loop; I kept my sides lesser than 100; most "economical" loop (I think): Loop a from 2 to 98 Loop b from a to 99 If even(a) then s = b else s = b + 1 (s = start with) f = a + b -1 (f = finish with) If f > 99 then f = 99 Loop c from s to f incrementing c by 2 ** ...go to work! ** this is made possible due to the "s" calculation; also results in a+b+c being even at all times. And this gives 5 "natives" (all sides < 100): a b c x y perimeters areas 5 5 6 2 6 16 12 13 20 21 6 21 54 126 25 51 52 12 52 128 624 53 53 56 21 60 162 1260 68 87 95 30 95 250 2850 BUT I didn't like the fact that in every case, there are at least 2 sides equal; so to add some dignity to this puzzle, I looked for the cases where NONE of the 5 sides were equal. And I found 10 "natives" (all sides < 500): a b c x y perimeters areas 37 130 157 12 150 324 1800 63 298 325 28 315 686 8820 81 113 130 36 126 324 4536 110 261 349 30 330 720 9900 132 265 353 45 330 750 14850 145 174 221 60 210 540 12600 149 245 390 14 378 784 5292 164 195 281 60 260 640 15600 175 208 303 63 280 686 17640 195 259 314 84 300 768 25200 684 745 821 285 840 2250 239400 ** ** in case someone wants the highest "a" case, where all sides < 1000 !!

Al Zimmermann points out this parametric solution, along with his solution:

The following is a solution for any positive integer m: (2m^2+2m+1, m^3+2m^2+2m, m^3+2m^2+2m+1) = (m^2+m, m^3+2m^2+2m+1) Notation: If a triangle with sides (a, b, c) and a rectangle with sides (x, y) are a solution, we'll write (a, b, c) = (x, y). If (a, b, c) = (x, y) is a solution, then we can form an infinite number of solutions by "scaling up" by any integral factor m. That is, (ma, mb, mc) = (mx, my) is also a solution. Let's call any solution which cannot be derived by scaling up a "primitive" solution. This is a list of all primitive solutions with perimeter less than 2500, listed in order of increasing perimeter: (5, 5, 6) = (2, 6), P = 16, A = 12 (13, 20, 21) = (6, 21), P = 54, A = 126 (25, 51, 52) = (12, 52), P = 128, A = 624 (53, 53, 56) = (21, 60), P = 162, A = 1260 (68, 87, 95) = (30, 95), P = 250, A = 2850 (41, 104, 105) = (20, 105), P = 250, A = 2100 (81, 113, 130) = (36, 126), P = 324, A = 4536 (37, 130, 157) = (12, 150), P = 324, A = 1800 (61, 185, 186) = (30, 186), P = 432, A = 5580 (145, 174, 221) = (60, 210), P = 540, A = 12600 (164, 195, 281) = (60, 260), P = 640, A = 15600 (195, 232, 259) = (84, 259), P = 686, A = 21756 (148, 265, 273) = (70, 273), P = 686, A = 19110 (85, 300, 301) = (42, 301), P = 686, A = 12642 (175, 208, 303) = (63, 280), P = 686, A = 17640 (63, 298, 325) = (28, 315), P = 686, A = 8820 (110, 261, 349) = (30, 330), P = 720, A = 9900 (132, 265, 353) = (45, 330), P = 750, A = 14850 (195, 259, 314) = (84, 300), P = 768, A = 25200 (149, 245, 390) = (14, 378), P = 784, A = 5292 (267, 365, 392) = (120, 392), P = 1024, A = 47040 (113, 455, 456) = (56, 456), P = 1024, A = 25536 (287, 401, 562) = (105, 520), P = 1250, A = 54600 (157, 481, 612) = (40, 585), P = 1250, A = 23400 (265, 471, 544) = (120, 520), P = 1280, A = 62400 (189, 500, 661) = (45, 630), P = 1350, A = 28350 (424, 485, 549) = (180, 549), P = 1458, A = 98820 (260, 595, 603) = (126, 603), P = 1458, A = 75978 (145, 656, 657) = (72, 657), P = 1458, A = 47304 (113, 630, 715) = (36, 693), P = 1458, A = 24948 (296, 625, 679) = (140, 660), P = 1600, A = 92400 (532, 533, 685) = (210, 665), P = 1750, A = 139650 (447, 763, 790) = (210, 790), P = 2000, A = 165900 (181, 909, 910) = (90, 910), P = 2000, A = 81900 (493, 565, 942) = (130, 870), P = 2000, A = 113100 (661, 663, 724) = (264, 760), P = 2048, A = 200640 (157, 916, 975) = (72, 952), P = 2048, A = 68544 (500, 543, 1015) = (63, 966), P = 2058, A = 60858 (684, 745, 821) = (285, 840), P = 2250, A = 239400

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