## An Equal Rectangle and Triangle

Find a rectangle and a triangle, with integer sides, which share both the same perimeter and area.

(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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