# proof of Apollonius theorem

Let $m$ be a median of the triangle, as shown in the figure.

By Stewart’s theorem we have

$$a\left({m}^{2}+{\left(\frac{a}{2}\right)}^{2}\right)={b}^{2}\left(\frac{a}{2}\right)+{c}^{2}\left(\frac{a}{2}\right)$$ |

and thus

$${m}^{2}+{\left(\frac{a}{2}\right)}^{2}=\frac{{b}^{2}+{c}^{2}}{2}.$$ |

Multiplying both sides by $2$ gives

$$2{m}^{2}+\frac{{a}^{2}}{2}={b}^{2}+{c}^{2}.$$ |

QED

Title | proof of Apollonius theorem^{} |

Canonical name | ProofOfApolloniusTheorem1 |

Date of creation | 2013-03-22 12:41:20 |

Last modified on | 2013-03-22 12:41:20 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 5 |

Author | drini (3) |

Entry type | Proof |

Classification | msc 51-00 |

Related topic | Median |

Related topic | Cevian |

Related topic | ApolloniusTheorem |

Related topic | StewartsTheorem |

Related topic | ProofOfStewartsTheorem |

Related topic | ProofOfApolloniusTheorem |