Pythagorean Triangle whose Area is Half Perimeter
Jump to navigation
Jump to search
Theorem
The $3-4-5$ triangle is the only Pythagorean triangle whose area is half its perimeter.
Proof
Let $a, b, c$ be the lengths of the sides of a Pythagorean triangle $T$.
Thus $a, b, c$ form a Pythagorean triple.
By definition of Pythagorean triple, $a, b, c$ are in the form:
- $2 m n, m^2 - n^2, m^2 + n^2$
We have that $m^2 + n^2$ is always the hypotenuse.
Thus the area of $T$ is given by:
- $\AA = m n \paren {m^2 - n^2}$
The perimeter of $T$ is given by:
- $\PP = m^2 - n^2 + 2 m n + m^2 + n^2 = 2 m^2 + 2 m n$
We need to find all $m$ and $n$ such that $\PP = 2 \AA$.
Thus:
\(\ds 2 m^2 + 2 m n\) | \(=\) | \(\ds 2 m n \paren {m^2 - n^2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds m + n\) | \(=\) | \(\ds n \paren {m + n} \paren {m - n}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \paren {m - n}\) | \(=\) | \(\ds 1\) |
As $m$ and $n$ are both (strictly) positive integers, it follows immediately that:
- $n = 1$
- $m - n = 1$
and so:
- $m = 2, n = 1$
and the result follows.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$