# isogonal conjugate

Let $\mathrm{\u25b3}ABC$ be a triangle, $AL$ the angle bisector^{} of $\mathrm{\angle}BAC$ and $AX$ any line passing through $A$. The isogonal conjugate^{} line to $AX$ is the line $AY$ obtained by reflecting the line $AX$ on the angle bisector $AL$.

In the picture $\mathrm{\angle}YAL=\mathrm{\angle}LAX$. This is the reason why $AX$ and $AY$ are called isogonal conjugates, since they form the same angle with $AL$. (iso= equal, gonal = angle).

Let $P$ be a point on the plane. The lines $AP,BP,CP$ are concurrent^{} by construction. Consider now their isogonals conjugates (reflections^{} on the inner angle bisectors). The isogonals conjugates will also concurr by the fundamental theorem on isogonal lines, and their intersection point $Q$ is called the isogonal conjugate of $P$.

If $Q$ is the isogonal conjugate of $P$, then $P$ is the isogonal conjugate of $Q$ so both are often referred as an isogonal conjugate pair.

An example of isogonal conjugate pair is found by looking at the centroid of the triangle and the Lemoine point.

Title | isogonal conjugate |
---|---|

Canonical name | IsogonalConjugate |

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

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

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 7 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 51-00 |

Related topic | Symmedian^{} |

Related topic | LemoinePoint |

Related topic | FundamentalTheoremOnIsogonalLines |

Defines | isogonal conjugate pair |

Defines | isogonal |