# Simson’s line

Let $ABC$ a triangle and $P$ a point on its circumcircle^{} (other than $A,B,C$).
Then the feet of the perpendiculars^{} drawn from P to the sides $AB,BC,CA$ (or their prolongations) are collinear^{}.

In the picture, the line passing through $U,V,W$ is a Simson line^{} for $\mathrm{\u25b3}ABC$.

An interesting result form the realm of analytic geometry^{} states that the envelope formed by Simson’s lines when P varies is a circular hypocycloid of three points.

Title | Simson’s line |
---|---|

Canonical name | SimsonsLine |

Date of creation | 2013-03-22 12:24:34 |

Last modified on | 2013-03-22 12:24:34 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 17 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 51-00 |

Related topic | Circumcircle |

Related topic | Triangle |