## 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)

 (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).