Altitude

The altitudes of a triangle are the Cevians that are perpendicular to the legs opposite . The three altitudes of any triangle are concurrent at the orthocenter (Durell 1928). This fundamental fact did not appear anywhere in Euclid's Elements.

The triangle connecting the feet of the altitudes is known as the orthic triangle.

The altitudes of a triangle with side length , , and and vertex angles , , have lengths given by

			(1)
			(2)
			(3)

where is the circumradius of . This leads to the beautiful formula

(4)

Other formulas satisfied by the altitude include

(5)

where is the inradius, and

			(6)
			(7)
			(8)

where are the exradii (Johnson 1929, p. 189). In addition,

			(9)
			(10)
			(11)

where is again the circumradius.

The points , , , and (and their permutations with respect to indices; left figure) all lie on a circle, as do the points , , , and (and their permutations with respect to indices; right figure).

Triangles and are inversely similar.

Additional properties involving the feet of the altitudes are given by Johnson (1929, pp. 261-262). The line joining the feet to two altitudes of a triangle is antiparallel to the third side (Johnson 1929, p. 172).

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