On with the story. My initial guess was that there would be an infinite
number of these triangles and it
seemed to me that the obvious way to attack this problem was to use
Heron's formula for the area of a triangle.
where are
the lengths of the three sides of the triangle and
. I let the sides of the triangle
be
represented by the three consecutive integers
. I then found, after simplification, that

(1) |

Upon examination of this result it was clear that had to be
an even integer in order for to be an integer.
For if were odd, would be odd and
even if were a perfect square, would not
divide . Also, by observation, it is clear that must be equal to a number of the form . I did
a quick search and found that the integers
all worked. Since my assumption was that there were going to be an
infinite number of these triangles, I looked for a recurrence relation
for which were the first three terms that
generated and infinite sequence of values for satisfying
(1). I would then test this sequence of to see if the values always
determined a Super-Heronian triangle.

Looking at the three terms, , my guess was that
these values satisfied the sequence given by the recurrence relation

(2) |

I used this sequence to generate more terms, all of which seemed to
work (see the table on the next page).

The sequence generated by the recurrence relation (2) can also be
represented by

(3) |

This sequence can be obtained from the auxiliary equation which has the roots . Therefore we let

Here are the first terms terms which do, in fact, generate
Super-Heronian triangles. The first row generates a line which
represents a degenerate triangle.

a | b | c | Area |

1 | 2 | 3 | 0 |

3 | 4 | 5 | 6 |

13 | 14 | 15 | 84 |

51 | 52 | 53 | 1170 |

193 | 194 | 195 | 16296 |

723 | 724 | 725 | 226974 |

2701 | 2702 | 2703 | 3161340 |

10083 | 10084 | 10085 | 44031786 |

37633 | 37634 | 37635 | 613283664 |

140451 | 140452 | 140453 | 8541939510 |

524173 | 524174 | 524175 | 118973869476 |

As one can see, the sides get large very fast. It is left to show that
every term of the sequence yields a solution;
that is, an area that is an integer. Let

Then

Now, by using mathematical induction, this last expression can be shown to be an integer for all integers . (Left as an exercise for the reader.) Hence, since the sequence is infinite, there are an infinite number of Super-Heronian triangles. It should be noted that we have not shown that our sequence generates all Super-Heronian triangles, but it does verify that there are an infinite number of them.

Let us examine another interesting feature of these triangles. Suppose
we use the area formula . We then have the area of the triangle given
by

where is the length of side and is the altitude to side . Thus,

which is an integer. Moreover, divides the triangle into two right triangles with side divided into two lengths, and . It is interesting to note that one of these right triangles has sides that are a primitive Pythagorean triple. In the table below, the asterisks mark the primitive triple in each row. Note how the primitve triples alternate.

1 | 2 | 3 | 0 | - | - |

3 | 4 | 5* | 3* | 0 | 4* |

13* | 14 | 15 | 12* | 5* | 9 |

51 | 52 | 53* | 45* | 24 | 28* |

193* | 194 | 195 | 168* | 95* | 99 |

723 | 724 | 725* | 627* | 360 | 364* |

2701* | 2702 | 2703 | 2340* | 1349* | 1353 |

10083 | 10084 | 10085* | 8733* | 5040 | 5044* |

37633* | 37634 | 37635 | 32592* | 18815* | 18819 |

140451 | 140452 | 140453* | 121635* | 70224 | 70228* |

524173* | 524174 | 524175 | 453948* | 262085* | 262089 |

One might observe that the s can be generated on
their own from the recurrence relation

One can investigate these triangles and find many interesting properties. I will leave further investigation to the reader.

This document was generated using the
**LaTeX**2`HTML`
translator Version 2002-2-1 (1.70)

Copyright © 1993, 1994, 1995, 1996,
Nikos Drakos,
Computer Based Learning Unit, University of Leeds.

Copyright © 1997, 1998, 1999,
Ross Moore,
Mathematics Department, Macquarie University, Sydney.

The command line arguments were:

**latex2html** `-white heronian.tex`

The translation was initiated by Bill Richardson on 2007-04-12

Bill Richardson 2007-04-12