# proof of parallelogram law

The proof follows directly from Apollonius theorem^{} noticing that each diagonal
is a median for the triangles in which parallelogram^{} is split by the other diagonal.
Also, the diagonals bisect each other.

Therefore, Apollonius theorem implies

$$2{\left(\frac{{d}_{1}}{2}\right)}^{2}+{\left(\frac{{d}_{2}}{2}\right)}^{2}={u}^{2}+{v}^{2}.$$ |

Multiplying both sides by $2$ and simplification leads to the desired expression.

Title | proof of parallelogram law^{} |
---|---|

Canonical name | ProofOfParallelogramLaw |

Date of creation | 2013-03-22 12:41:29 |

Last modified on | 2013-03-22 12:41:29 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 5 |

Author | Mathprof (13753) |

Entry type | Proof |

Classification | msc 51-00 |

Related topic | ApolloniusTheorem |

Related topic | Median |

Related topic | ProofOfParallelogramLaw2 |