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).
See also Cevian, Foot, Orthocenter, Perpendicular, Perpendicular Foot
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 Eric W. Weisstein