Consider a triangle abc as shown.
Heron's Theorem states that the area of a triangle with sides a, b, and c can be determined using the following formula:
where s is the semi-perimeter of the triangle, which is equal
to 
.
Heron's theorem can be developed by considering a triangle abc with an altitude h and then applying the Pythagorean theorem. To see a proof of this theorem, click here.
The theorem allows one to calculate the areas for any triangles whose sides are known. This tool can be useful for exploring some characteristics of triangles.
Consider the following story problem from a mathematical puzzle book:
"I wish I could keep my grass green like yours," said Bob, leaning over the low fence as his neighbor stopped the mower. "But why the new flower beds, all triangular and different?"
"Just an idea of mine." Professor Brayne chuckled. "The distance around each bed in feet is the same as its area in square feet, and the sides are all whole numbers in feet."
Bob pondered this a moment. "I guess that took some figuring out. How many more triangles do you plan to have just like that, still all different?"
The professor smiled. "I can't have any more," he replied. "You can see for yourself."
There was plenty of space for many more on that great expanse of lawn, but Bob wasn't arguing.
How many of those special flower beds were there, and what were their dimensions?
In order to answer the problem, we are led to the search for perfect triangles. For the purpose of this essay a perfect triangle will be defined as any triangle having side lengths that are integers and for which the area and perimeter of the triangle are equal.
Using a hypothetical triangle with side lengths of x + y, y + z, and x + z, the problem can be narrowed down algebraically. For this triangle, the perimeter is equal to
which equals 
and the semiperimeter
is 
.
Substituting into the equation for the area given by Heron's Theorem, we have that the area equals:
which simplifies to
that must equal the perimeter of for any perfect triangle.
Squaring both sides yields the following:
which then simplifies to the equation
.Rearranging yields the equation
In order to solve the equation we establish the following inequality for the arbitrary values of x, y, and z:
Since x, y, and z must be whole numbers, choosing values for the smallest variable can determine the possible solutions. When z = 1, the equation then simplifies to the following:
Using the inequality x > y > z, different y values are used to find possible solutions. This process quickly shows that there are no positive values for x until y > 4. For y = 5, x = 24 solves the equation. Because the sides of the triangle are x + y, x + z, and y + z, this solution yields a triangle with sides of 6, 25, and 29. For y = 6, x = 14 and the sides of the triangle are 7, 15, and 20. For y = 7 there is no integral solution for x. For y = 8, x = 9 and the sides of the triangles are 9, 10, and 17. For values of y greater than 8, the value of x is less than y which contradicts the inequality x > y > z. Therefore these three solutions are the only ones for z = 1.
Repeating the process above, next we try z = 2 and find that y > 2 is necessary to obtain positive values of x. For y = 3, x = 10 and the sides of the triangle are 5, 12, and 13 - a familiar right triangle. Next for y = 4, x = 6 and the sides of the triangle are 6, 8, and 10 - another familiar right triangle! For values of y greater than 4, solutions of x are less than y which contradicts the inequality x > y > z.
Now for z = 3. In order to obtain positive values of x, y only needs to be greater than 1, however the inequality x > y > z necessitates that y > 3. For z = 3 and y = 3, there is no integral solution for x. For y = 4, there is no integral solution for x and the rational solution of x is less than 4. Therefore as y continues to increase, x will decrease and since x > y, there are no solutions for z = 3.
When z = 4, the minimum value for y is by the inequality is 4. There are no integral solutions for x when y = 4 and the rational solution of x is less than 4. This means that there are no possible solutions for z = 4 or for any value of z greater than 4.
So there are only five perfect triangles found in Professor Brayne's garden. The table below gives the side lengths for all five perfect triangles:
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Another approach to finding these perfect triangles involves using a Microsoft Excel spreadsheet to systematically investigate the areas of triangles with integer perimeters using Heron's formula and the Triangle Inequality Theorem. The Triangle Inequality theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side. Using this information, a spreadsheet can be developed to determine the areas of triangles with integer perimeters and integer side measures. The columns in the spreadsheet are illustrated below:
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Semiperimeter |
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Since the smallest possible value for the perimeter of a triangle with integer side lengths is 3, this is the first entry in the spreadsheet. The perimeter is systematically increased and the area evaluated to find those triangles for which the area is an integer and equal to the perimeter. The first twenty rows of the spreasheet are shown below:
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Semiperimeter |
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As shown in the table (and determined from the algebra earlier), there are no perfect triangles with perimeter less than or equal to 12. There is only one triangle in the twenty rows that has an integer area, and it happens to be the 3, 4, 5 right triangle.
The spreadsheet can be used to continually investigate these triangles, searching for the perfect triangles. Although it takes many rows to find all five of these triangles, the spreadsheet can perform the calculations with Heron's formula rapidly and the systematic investigation of all the possibilities leads to some interesting observations and conjectures.
For the purpose of the following discussion, I will define the term "Integral Triangle" to be any triangle which has integer side lengths and an integer area. While the five perfect triangles discovered previously certainly fall under this category, there are infinitely more triangles that meet the criteria for the definition of the "Integral Triangle".
In exploring the perfect triangles using an Excel spreadsheet, all the integral triangles along the way were easy to determine. Although the difference between the area and the perimeter in the final column of the spreadsheet was not zero, it was an integral value, as was the value of the area in the next to last column. The first of these integral trianlges was found in the first twenty rows of the spreadsheet, the 3-4-5 right triangle. Further exploration yielded many more integral triangles as the possibilities were investigated for all triangles having a perimeter less than or equal to 100. As the investigation continued, certain patterns emerged that led to some interesting conjectures.
The see the table containing all of the integral triangles that have a perimeter less than or equal to 100, click here.
The collection of integral triangles leads to some propositions:
(Click here for further explanation)
(Click here for further explanation)
(Click here for further explanation)
(Click here for further explanation)
(Click here for further explanation)