Eisenstein triple
Similar to a Pythagorean triple, an Eisenstein triple is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees.
Triangles with an angle of 60°
An Eisenstein triple
Triangles with an angle of 60° are a special case of the Law of Cosines:[1][2][3]
When the lengths of the sides are integers, the values form a set known as an Eisenstein triple.[4]
Examples of Eisenstein triples include:[5]
| Side a | Side b | Side c |
|---|---|---|
| 3 | 8 | 7 |
| 5 | 8 | 7 |
| 5 | 21 | 19 |
| 7 | 15 | 13 |
| 7 | 40 | 37 |
| 8 | 15 | 13 |
| 9 | 24 | 21 |
Triangles with an angle of 120°
Triangle with 120° angle and integer sides
A similar special case of the Law of Cosines relates the sides of a triangle with an angle of 120 degrees:
Examples of such triangles include:[6]
| Side a | Side b | Side c |
|---|---|---|
| 3 | 5 | 7 |
| 7 | 8 | 13 |
| 5 | 16 | 19 |
References
- Gilder, J., Integer-sided triangles with an angle of 60°," Mathematical Gazette 66, December 1982, 261 266
- Burn, Bob, "Triangles with a 60° angle and sides of integer length," Mathematical Gazette 87, March 2003, 148–153.
- Read, Emrys, "On integer-sided triangles containing angles of 120° or 60°", Mathematical Gazette, 90, July 2006, 299–305.
- "Archived copy" (PDF). Archived from the original (PDF) on 2006-07-23. Retrieved 2014-05-05.CS1 maint: archived copy as title (link)
- http://www.had2know.com/academics/integer-triangles-60-degree-angle.html
- http://www.had2know.com/academics/integer-triangles-120-degree-angle.html
External links
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