Triangle Perimeters and Areas

Similar to the previous rectangle problem...
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.
There are actually many solutions for the questions above, so try to find the triangles with the minimum perimeters. I'd be very interested to know of any formulas used by people to obtain their solutions.

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

Solutions were received from Philippe Fondanaiche, Le Ba Nguyen, Ravi Subramanian, Kirk Bresniker, Al Zimmermann, and Sandy Thompson. The smallest-perimeter solutions are:
```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.
```
Every solver used Heron's formula:
• s = (a+b+c)/2
• A = sqrt[s(s-a)(s-b)(s-c)]

Mail to Ken
Update 11/15/99: Oliver Stone solved this puzzle in a very thorough way, so I thought I'd include his solution. It's a Word 97 file: Oliver's Stone's Solution.
Al Zimmermann's exhaustive list of primitive solutions follows:
```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)
```