Triangles with Integer Area and Integer Sides in Arithmetical Sequence
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Theorem
The triangles with the following sides in arithmetic sequence have integer areas:
- $3, 4, 5$
- $13, 14, 15$
- $15, 28, 41$
- $15, 26, 37$
Their areas are:
- $6, 84, 126, 156$
Proof
From Heron's Formula, the area $A$ of $\triangle ABC$ is given by:
- $A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac{a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
For $3, 4, 5$:
\(\ds s\) | \(=\) | \(\ds \frac {3 + 4 + 5} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds \sqrt {6 \paren {6 - 3} \paren {6 - 4} \paren {6 - 5} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {6 \times 3 \times 2 \times 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {6 \times 6}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6\) |
For $13, 14, 15$:
\(\ds s\) | \(=\) | \(\ds \frac {13 + 14 + 15} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds \sqrt {21 \paren {21 - 13} \paren {21 - 14} \paren {21 - 15} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {21 \times 8 \times 7 \times 6}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\paren {3 \times 7} \times 2^3 \times 7 \times \paren {2 \times 3} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 7 \times 2^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 84\) |
For $15, 28, 41$:
\(\ds s\) | \(=\) | \(\ds \frac {15 + 28 + 41} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 42\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds \sqrt {42 \paren {42 - 15} \paren {42 - 28} \paren {42 - 41} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {42 \times 27 \times 14 \times 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\paren {2 \times 3 \times 7} \times 3^3 \times \paren {2 \times 7} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \times 3^2 \times 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 126\) |
For $15, 26, 37$:
\(\ds s\) | \(=\) | \(\ds \frac {15 + 26 + 37} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 39\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds \sqrt {39 \paren {39 - 15} \paren {39 - 26} \paren {39 - 37} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {39 \times 24 \times 13 \times 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\paren {3 \times 13} \times \paren {2^3 \times 3} \times 13 \times 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^2 \times 3 \times 13\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 156\) |
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Sources
- 1989: John MacNeill: 13, 14, 15 and 15, 26, 37 (Mathematical Spectrum Vol. 21, no. 3: pp. 83 – 84)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $126$