The altitudes of a Triangle are the Cevians which are Perpendicular to the
Legs opposite . They have lengths
given by

(1) |

(2) |

(3) |

Other formulas satisfied by the altitude include

(4) |

(5) |

(6) |

(7) |

(8) |

The points , , , and (and their permutations with respect to indices) all lie on a Circle, as do the points , , , and (and their permutations with respect to indices). Triangles and are inversely similar.

The triangle has the minimum Perimeter of any Triangle inscribed in a given Acute Triangle (Johnson 1929, pp. 161-165). The Perimeter of is (Johnson 1929, p. 191). Additional properties involving the Feet of the altitudes are given by Johnson (1929, pp. 261-262).

**References**

Coxeter, H. S. M. and Greitzer, S. L. *Geometry Revisited.* Washington, DC: Math. Assoc. Amer., pp. 9 and 36-40, 1967.

Johnson, R. A. *Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.*
Boston, MA: Houghton Mifflin, 1929.

© 1996-9

1999-05-25