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 |
See also[]
- Integer triangles with a 60° angle
- Integer triangles with a 120° angle
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.
{{cite web}}: CS1 maint: archived copy as title (link) - ^ "Integer Triangles with a 60-Degree Angle".
- ^ "Integer Triangles with a 120-Degree Angle".
External links[]
Categories:
- Arithmetic problems of plane geometry
- Triangle geometry
- Diophantine equations