Smallest: [65, 70, 75] P = 210, A = 2100

Largest:  [73, 14606, 14667] P = 29,346 and A = 293,460  (at least that I found)

This is a very thin triangle with the smallest angle being about 9.42 minutes of arc.

The smallest of the group will be the closest to equilateral and as they get larger, they get thinner.

I tried to find a theoretical maximum area, thinnest possible triangle that would have an A/P ratio of 10, but I did not come to a solid finding.  But I did notice some things about the side lengths that made it possible to put more strict constraints on the search parameters.

Call the sides x,y,z where x <= y <= z.

As the triangles get larger and thinner, y gets larger and also x+y-z gets smaller. 

The ratio (x+y-z)/y gets very small, e.g. 0.01

----------------------------

Version one:

big = 10000

for x in range(1,big):

    for y in range(x,big):

        lo = min(1, y-x)

        hi = x+y

        for z in range(lo,hi):

            p = hi + z

            a = areaHeron(x,y,z)

            if a/p == 10:

                tri = sorted([x,y,z])

                if [int(a),tri] not in solutions:

                    solutions.append([int(a),tri])

print(sorted(solutions))

-----

Version two squeezing down on the parameters for x,y,z

solutions = []

big = 1000000

for x in range(40,90):

    print(2*x - 80, '%') # to follow progress

    for y in range(10000,big):

        lo = x + int(999*y/1000)

        hi = x+y

        for z in range(lo,hi):

            p = hi + z

            a = areaHeron(x,y,z)

            if a/p == 10:

                tri = sorted([x,y,z])

                if [int(a),tri] not in solutions:

                    solutions.append([int(a),tri])

                    print([int(a),tri])

print(sorted(solutions))

print('number of solutions', len(solutions))

-----  -----

Output of version one with "big" variable 10000

   AREA    SIDES

  [2100, [65, 70, 75]],

  [2160, [60, 78, 78]],

  [2220, [61, 74, 87]],

  [2280, [57, 82, 89]],

  [2340, [65, 72, 97]],

  [2400, [60, 80, 100]],

  [2520, [56, 90, 106]],

  [2640, [52, 102, 110]],

  [2640, [66, 82, 116]],

  [2700, [75, 75, 120]],

  [2940, [49, 122, 123]],

  [3000, [50, 120, 130]],

  [3060, [53, 117, 136]],

  [3240, [72, 102, 150]],

  [3300, [55, 125, 150]],

  [3360, [48, 140, 148]],

  [3360, [68, 112, 156]],

  [3600, [90, 100, 170]],

  [3720, [62, 136, 174]],

  [3780, [54, 149, 175]],

  [3900, [75, 130, 185]],

  [4200, [50, 175, 195]],

  [4200, [70, 150, 200]],

  [4320, [108, 116, 208]],

  [4500, [45, 200, 205]],

  [4620, [56, 187, 219]],

  [4620, [66, 175, 221]],

  [4680, [74, 169, 225]],

  [4740, [87, 158, 229]],

  [4980, [83, 174, 241]],

  [5280, [44, 240, 244]],

  [5460, [78, 203, 265]],

  [5520, [46, 246, 260]],

  [5640, [47, 250, 267]],

  [6000, [60, 250, 290]],

  [6180, [109, 206, 303]],

  [6240, [52, 272, 300]],

  [6300, [65, 259, 306]],

  [6360, [106, 218, 312]],

  [6600, [165, 170, 325]],

  [6660, [153, 185, 328]],

  [6960, [58, 300, 338]],

  [7020, [51, 312, 339]],

  [7020, [135, 221, 346]],

  [7500, [125, 255, 370]],

  [7740, [43, 362, 369]],

  [7980, [119, 285, 394]],

  [8100, [45, 375, 390]],

  [8160, [68, 348, 400]],

  [8160, [204, 208, 404]],

  [8280, [184, 234, 410]],

  [8580, [169, 264, 425]],

  [9120, [156, 304, 452]],

  [9240, [42, 440, 442]],

  [9240, [110, 357, 457]],

  [9240, [154, 312, 458]],

  [9900, [55, 450, 485]],

  [10080, [144, 364, 500]],

  [10320, [86, 436, 510]],

  [10500, [105, 425, 520]],

  [10620, [59, 481, 522]],

  [11220, [51, 521, 550]],

  [11220, [136, 429, 557]],

  [11340, [42, 543, 549]],

  [11640, [102, 485, 577]],

  [11880, [54, 550, 584]],

  [12120, [101, 510, 601]],

  [12180, [203, 409, 606]],

  [12600, [130, 504, 626]],

  [12900, [129, 520, 641]],

  [12960, [48, 612, 636]],

  [13860, [63, 638, 685]],

  [14280, [42, 689, 697]],

  [14580, [81, 654, 723]],

  [14880, [124, 624, 740]],

  [16080, [402, 404, 802]],

  [16740, [62, 783, 829]],

  [17100, [95, 765, 850]],

  [17220, [41, 840, 841]],

  [17400, [120, 754, 866]],

  [18060, [43, 875, 888]],

  [18060, [303, 602, 901]],

  [18120, [302, 606, 904]],

  [18480, [44, 894, 910]],

  [19560, [163, 818, 975]],

  [19740, [47, 952, 975]],

  [19740, [282, 707, 985]],

  [20340, [117, 904, 1013]],

  [20460, [93, 935, 1018]],

  [21060, [78, 981, 1047]],

  [32040, [801, 802, 1601]]

-----  -----

Output of version two testing x from 40 to 89; y 10000 to 1000000; "lo" factor 999/1000

(this misses many smaller triangles, but is looking for the largest 

 [28140, [67, 1347, 1400]],

 [30600, [90, 1445, 1525]],

 [31200, [60, 1508, 1552]],

 [31920, [76, 1526, 1590]],

 [37380, [89, 1785, 1864]],

 [41820, [51, 2050, 2081]],

 [46920, [46, 2312, 2334]],

 [60180, [59, 2958, 3001]],

 [68040, [42, 3375, 3387]],

 [71340, [87, 3485, 3562]],

 [75480, [74, 3706, 3768]],

 [87480, [54, 4329, 4365]],

 [139320, [86, 6885, 6961]],