# Integer Triangles with a 60° Angle

Just as there are integer *right* triangles, so there are integer 60° triangles. In such triangles, the side lengths are all integers and one of the angles measures exactly 60°.

If **a** and **b** denote the two sides which are adjacent to the 60°, and **c** denotes the remaining side, then **a**, **b**, and **c** satisfy a Pythagorean-like relation:

**a² - ab + b² = c²**.

This relation holds for all 60° triangles, not just those with integer sides.

To find triples (a, b, c) that satisfy the relation a² - ab + b² = c², you can use generator functions similar to the generator functions for Pythagorean triples and 120° triangles, or you can use a trick based on knowledge of 120° triangles. To produce 60° triangles with generator functions, the equations are

a = m² - n²

b = 2mn - n²

c = m² - mn + n²,

where m and n are integers and m > n. This works because

(m² - n²)² - (m² - n²)(2mn - n²) + (2mn - n²)² = (m² - mn + n²)²

### How to Generate 60° Triangles With 120° Triangles

Suppose the triple (x, y, z) represents an integer 120° triangle, i.e., x² + xy + y² = z². Then the triples (x+y, y, z) and (x, x+y, z) represent two integer 60° triangles. To confirm this algebraically, start with the known relation x² + xy + y² = z². Rewrite the left hand side in an equivalent form and simplify:x² + xy + y² = z²

x² - x² + x² + xy - xy + xy + y² = z²

x² + 2xy + y² - x² - xy + x² = z²

(x+y)² - (x+y)x + x² = z²

This shows that (x, x+y, z) is an integer 60° triangle. A similar argument shows that (x+y, y, z) is as well. For visual proof, see the animated image below:

You can use this trick to generate every primitive integer 60° triangle from every primitive integer 120° triangle. The table below shows the first few primitive triples (a, b, c) where c < 100.

a | b | c |

3 | 8 | 7 |

5 | 8 | 7 |

7 | 15 | 13 |

8 | 15 | 13 |

5 | 21 | 19 |

16 | 21 | 19 |

11 | 35 | 31 |

24 | 35 | 31 |

7 | 40 | 37 |

33 | 40 | 37 |

13 | 48 | 43 |

35 | 48 | 43 |

16 | 55 | 49 |

39 | 55 | 49 |

9 | 65 | 61 |

56 | 65 | 61 |

32 | 77 | 67 |

45 | 77 | 67 |

17 | 80 | 73 |

63 | 80 | 73 |

40 | 91 | 79 |

51 | 91 | 79 |

11 | 96 | 91 |

85 | 96 | 91 |

19 | 99 | 91 |

80 | 99 | 91 |

55 | 112 | 97 |

57 | 112 | 97 |

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