# Integer Triangles with a 120° Angle

Just as there are integer *right* triangles, so there are integer 120° triangles. In such triangles, all side lengths are whole numbers and the obtuse angle measures exactly 120°.

If the two sides adjacent to the 120° angle are **a** and **b** and the third side is **c**, then **a**, **b**, and **c** satisfy a Pythagorean-like relation:

**a² + ab + b² = c²**,

which holds for all 120° triangles, not just the integer ones.

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. In the 120° case, the equations are

a = m² - n²

b = 2mn + n²

c = m² + mn + n²,

where m and n are integers and m > n. Observe:

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

In this way you can come up with infinitely many triples (a, b, c). The first few primitive triples (those in which a, b, and c have no factors in common) are given in the table below:

a | b | c |

3 | 5 | 7 |

7 | 8 | 13 |

5 | 16 | 19 |

11 | 24 | 31 |

7 | 33 | 37 |

13 | 35 | 43 |

16 | 39 | 49 |

9 | 56 | 61 |

32 | 45 | 67 |

17 | 63 | 73 |

40 | 51 | 79 |

11 | 85 | 91 |

19 | 80 | 91 |

55 | 57 | 97 |

The first few triples with consecutive integer side lengths are

(7, 8, 13)

(104, 105, 181)

(1455, 1456, 2521)

(20272, 20273, 35113).

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