Heronian Triangle
A Heronian triangle is a triangle having rational side lengths and rational area. The triangles are so named because such triangles are related to Heron's formula
 |
(1)
|
giving a triangle area 
in terms of
its side lengths 
, 
, 
and semiperimeter 
. Finding a Heronian triangle is therefore
equivalent to solving the Diophantine equation
 |
(2)
|
The complete set of solutions for integer Heronian triangles (the three side lengths and area can be multiplied by their least common multiple to make them all integers) were found by Euler (Buchholz 1992; Dickson 2005, p. 193), and parametric versions were given by Brahmagupta and Carmichael (1952) as
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
This produces one member of each similarity class of Heronian triangles for any integers 
, 
, and 
such that 
, 
,
and 
(Buchholz 1992).
The first few integer Heronian triangles sorted by increasing maximal side lengths, are ((3, 4, 5), (5, 5, 6), (5, 5, 8), (6, 8, 10), (10, 10, 12), (5, 12, 13), (10,
13, 13), (9, 12, 15), (4, 13, 15), (13, 14, 15), (10, 10, 16), ... (OEIS A055594,
A055593, and A055592),
having areas 6, 12, 12, 24, 48, 30, 60, 54, ... (OEIS A055595).
The first few integer Heronian scalene triangles,
sorted by increasing maximal side lengths, are (3, 4, 5), (6, 8, 10), (5, 12, 13),
(9, 12, 15), (4, 13, 15), (13, 14, 15), (9, 10, 17), ... (OEIS A046128,
A046129, and A046130),
having areas 6, 24, 30, 54, 24, 84, 36, ... (OEIS A046131).
R. Rathbun has cataloged all integer Heronian triangles with perimeters smaller
than 
(Peterson 2003).
Schubert (1905) claimed that Heronian triangles with two rational triangle medians do not exist (Dickson 2005). This was shown to be incorrect by Buchholz
and Rathbun (1997), who discovered the triangles given in the following table, where
are triangle
median lengths and 
is the area.
 |  |  |  |  |  |
| 73 | 51 | 26 |  |  | 420 |
| 626 | 875 | 291 | 572 |  | 55440 |
| 4368 | 1241 | 3673 | 1657 |  | 2042040 |
| 14791 | 14384 | 11257 |  | 11001 | 75698280 |
| 28779 | 13816 | 15155 |  | 21937 | 23931600 |
| 1823675 | 185629 | 1930456 |  |  | 142334216640 |
D. Borris (pers. comm., Oct. 22, 2003) considered primitive pairs of Heronian triangles, one a right triangle with sides 
and the
other an isosceles triangle with sides 
, such that the two triangle share a common
area and a common perimeter. Borris found the pair 
and 
(corresponding
to area 
and perimeter 864), with no other
such pairs with right triangle smallest side length less than 
.
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