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).
