# alternative proof of necessity direction of equivalent conditions for triangles (hyperbolic and spherical)

The following is a proof that, in hyperbolic geometry and spherical geometry, an equiangular triangle $\mathrm{\u25b3}ABC$ is automatically equilateral (http://planetmath.org/EquilateralTriangle) (and therefore regular^{} (http://planetmath.org/RegularTriangle)). It better the proof of sufficiency supplied in the entry equivalent conditions for triangles and is slightly shorter than the proof of necessity supplied in the same entry.

###### Proof.

Assume that $\mathrm{\u25b3}ABC$ is equiangular.

Since $\mathrm{\angle}A\cong \mathrm{\angle}B\cong \mathrm{\angle}C$, AAA yields that $\mathrm{\u25b3}ABC\cong \mathrm{\u25b3}BCA$. By CPCTC, $\overline{AB}\cong \overline{AC}\cong \overline{BC}$. Hence, $\mathrm{\u25b3}ABC$ is equilateral.

∎

Title | alternative proof of necessity direction of equivalent conditions for triangles (hyperbolic and spherical) |
---|---|

Canonical name | AlternativeProofOfNecessityDirectionOfEquivalentConditionsForTriangleshyperbolicAndSpherical |

Date of creation | 2013-03-22 17:12:55 |

Last modified on | 2013-03-22 17:12:55 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 5 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 51-00 |