## Altitude

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

 (1)

 (2)

where is the Semiperimeter of and . Another interesting Formula is
 (3)

(Johnson 1929, p. 191), where is the Area of the Triangle and is the Semiperimeter of the altitude triangle . The three altitudes of any Triangle are Concurrent at the Orthocenter . This fundamental fact did not appear anywhere in Euclid's Elements.

Other formulas satisfied by the altitude include

 (4)

 (5)

 (6)

where is the Inradius and are the Exradii (Johnson 1929, p. 189). In addition,
 (7)

 (8)

where is the Circumradius.

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
1999-05-25