Triangle Perimeters and Areas

1. Find the dimensions of all triangles that have area equal numerically to M times the perimeter, where all sides are whole numbers. (M=1,2,3,4,5)
2. Repeat for perimeter equal to M times the area.

Source: Extended from a puzzle submitted by Ravi Subramanian (where M=1).

M=1: (6,  8,  10); Perimeter = 24,  Area = 24
M=2: (13, 14, 15); Perimeter = 42,  Area = 84
M=3: (20, 20, 24); Perimeter = 64,  Area = 192
M=4: (26, 28, 30); Perimeter = 84,  Area = 336
M=5: (30, 39, 39); Perimeter = 108, Area = 540

M=1/2: (3, 4,  5); Area = 6,  Perimeter = 12 [This is the only solution.]
M=1/3: None.
M=1/4: None.
M=1/5: None.

• s = (a+b+c)/2
• A = sqrt[s(s-a)(s-b)(s-c)]

I've only attempted the first half of the problem, in which we seek triangles 
whose areas are numerically equal to m times their perimeters.

As in the previous rectangle problem, if a triagle (a,b,c) is a solution for 
m, then (ka,kb,kc) is a solution for km.  Call a solution "primitive" if it 
cannot be so dervived from a solution for a smaller m (that is, if the 
greatest common divisor of (a,b,c,m) is 1).

At the bottom of this is a list of all primitive solutions for m = 1 thru 5.

The following formulae work:
(4m+1, 8m^2+4m, 8m^2+4m+1) [always a right triangle]
(8m^2+1, 8m^2+2, 16m^2+1)
(4m^2+2, 4m^2+4, 8m^2+2) [not primitive if m is even]

The very last triangle for a given m seems always to be:
(4m^2+2, 16m^4+8m^2+1, 16m^4+12m^2+1)

And then there's these:
(4m^2+3, 8m^4+6m^2+1, 8m^4+10m^2+2)
(4m^2+5, 4m^4+5m^2+1, 4m^4+9m^2+4)

Note that some solutions, such as (9,10,17) for m=1, can be derived from more 
than one formula.

Specific solutions:
1: (6,8,10)
1: (5,12,13)
1: (9,10,17)
1: (7,15,20)
1: (6,25,29)

2: (13,14,15)
2: (15,15,24)
2: (11,25,30)
2: (15,26,37)
2: (10,35,39)
2: (9,40,41)
2: (33,34,65)
2: (25,51,74)
2: (9,75,78)
2: (11,90,97)
2: (21,85,104)
2: (19,153,170)
2: (18,289,305)

3: (20,20,24)
3: (17,25,26)
3: (17,25,28)
3: (20,21,29)
3: (16,30,34)
3: (15,34,35)
3: (22,26,40)
3: (14,48,50)
3: (25,33,52)
3: (24,35,53)
3: (16,52,60)
3: (20,51,65)
3: (14,61,65)
3: (19,60,73)
3: (38,40,74)
3: (35,44,75)
3: (32,50,78)
3: (30,56,82)
3: (29,60,85)
3: (13,84,85)
3: (28,65,89)
3: (26,80,102)
3: (25,92,113)
3: (17,105,116)
3: (13,122,125)
3: (24,110,130)
3: (14,130,136)
3: (73,74,145)
3: (23,140,159)
3: (55,111,164)
3: (49,148,195)
3: (16,195,205)
3: (22,200,218)
3: (46,185,229)
3: (43,259,300)
3: (21,380,397)
3: (41,370,409)
3: (40,481,519)
3: (39,703,740)
3: (38,1369,1405)

4: (25,29,36)
4: (25,38,51)
4: (21,89,100)
4: (25,84,101)
4: (35,73,102)
4: (51,98,145)
4: (17,144,145)
4: (27,219,240)
4: (129,130,257)
4: (117,145,260)
4: (105,169,272)
4: (97,195,290)
4: (91,225,314)
4: (21,340,353)
4: (85,273,356)
4: (81,325,404)
4: (78,385,461)
4: (75,481,554)
4: (17,550,555)
4: (73,585,656)
4: (25,803,822)
4: (70,897,965)
4: (35,1122,1153)
4: (18,1157,1165)
4: (69,1105,1172)
4: (67,2145,2210)
4: (66,4225,4289)

5: (30,39,39)
5: (28,45,53)
5: (26,51,55)
5: (33,41,58)
5: (24,70,74)
5: (36,51,75)
5: (34,56,78)
5: (31,68,87)
5: (54,58,104)
5: (22,120,122)
5: (23,123,130)
5: (26,136,150)
5: (53,109,156)
5: (29,150,169)
5: (34,174,200)
5: (102,104,202)
5: (92,117,205)
5: (21,220,221)
5: (78,152,226)
5: (77,156,229)
5: (72,182,250)
5: (43,218,255)
5: (27,275,292)
5: (65,252,313)
5: (24,306,318)
5: (62,312,370)
5: (201,202,401)
5: (60,377,433)
5: (151,303,452)
5: (22,447,455)
5: (21,533,538)
5: (26,525,541)
5: (57,572,625)
5: (126,505,629)
5: (121,606,725)
5: (56,702,754)
5: (30,754,776)
5: (38,763,795)
5: (23,1156,1167)
5: (111,1111,1220)
5: (54,1352,1402)
5: (37,1853,1884)
5: (106,2121,2225)
5: (53,2652,2701)
5: (105,2626,2729)
5: (103,5151,5252)
5: (102,10201,10301)