# fundamental theorem on isogonal lines

Let $\mathrm{\u25b3}ABC$ be a triangle^{} and $AX,BY,CZ$ three concurrent lines at $P$.
If $A{X}^{\prime},B{Y}^{\prime},C{Z}^{\prime}$ are the respective isogonal conjugate^{} lines for $AX,BY,CZ$, then $A{X}^{\prime},B{Y}^{\prime},C{Z}^{\prime}$ are also concurrent^{} at some point ${P}^{\prime}$.

An applications of this theorem proves the existence of Lemoine point (for it is the intersection^{} point of the symmedians^{}):

This theorem is a direct consequence of Ceva’s theorem (trigonometric version).

Title | fundamental theorem on isogonal lines |
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Canonical name | FundamentalTheoremOnIsogonalLines |

Date of creation | 2013-03-22 13:01:16 |

Last modified on | 2013-03-22 13:01:16 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 4 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 51-00 |

Related topic | Isogonal |

Related topic | IsogonalConjugate |

Related topic | LemoinePoint |

Related topic | Symmedian |

Related topic | Triangle |