Exploring Pythagorean Triangles with Equal Area and Perimeter

Pythagorean triangles, also known as right-angled triangles, are an interesting subject in mathematics, particularly in the field of number theory. A special case of these triangles is where the area of the triangle equals its perimeter, a topic of intrigue and exploration in the realm of geometry and number theory.

Primitive Pythagorean Triangles

A primitive Pythagorean triangle is one where the sides (a), (b), and (c) are coprime integers, and (c) is the hypotenuse. The sides can be represented by the formulas:

[a m^2 - n^2]

[b 2mn]

[c m^2 n^2]

where (m) and (n) are coprime integers, one of which is even, and both are greater than zero. This representation ensures that the sides are in their simplest form without any common factor other than 1.

Area and Perimeter Equivalence

The area (A) of the triangle is given by:

[A frac{1}{2}ab frac{1}{2}(m^2 - n^2)(2mn) mn(m^2 - n^2)]

The perimeter (P) of the triangle is the sum of the sides:

[P a b c (m^2 - n^2) 2mn (m^2 n^2) 2m^2 2mn]

To find the conditions where the area equals the perimeter, we set the two expressions equal:

[mn(m^2 - n^2) 2m(m n)]

Rearranging this equation, we get:

[mn^3 - mn^2 - 2m^2 - 2mn 0]

Factoring out (m), we simplify to:

[m(n^3 - n^2 - 2m - 2n) 0]

Since (m eq 0), we have the quadratic equation:

[m^2n - 2m - n^3 - 2n 0]

Using the quadratic formula, we solve for (m):

[m frac{2 pm sqrt{4 4n(n^3 2n)}}{2n} frac{1 pm sqrt{1 n^3 2n}}{n}]

For (m) to be an integer, (1 n^3 2n) must be a perfect square. Let:

[k^2 1 n^3 2n]

Testing for small values of (n):

[n 1] (k^2 1 1^3 2(1) 4), implying (k 2) (m frac{1 2}{1} 3), valid triangle sides: (a 8), (b 6), (c 10)

The area and perimeter are both 24, confirming that there is exactly one primitive Pythagorean triangle where the area equals the perimeter.

Conclusion

Through detailed mathematical analysis, we have shown that there is exactly one primitive Pythagorean triangle where the area equals the perimeter. This triangle has sides (a 8), (b 6), and (c 10), yielding an area and perimeter of 24. This finding adds depth to our understanding of Pythagorean triangles and their unique properties.