Given a triangle 
, the triangle 
whose vertices are endpoints of the altitudes
from each of the vertices of 
is called the orthic triangle, or sometimes the altitude
triangle. The three lines 
, 
, and 
are concurrent at the orthocenter 
of 
.
The orthic triangle is therefore both the pedal triangle and Cevian triangle with respect to 
(Kimberling 1998, p. 156). It is also the cyclocevian
triangle of the triangle centroid 
.
Its trilinear vertex matrix is
![[0 secB secC; secA 0 secC; secA secB 0].](/images/equations/OrthicTriangle/NumberedEquation1.svg) |
(1)
|
The area of the orthic triangle is given by
 |
(2)
|
where 
is the circumradius of 
.
The orthic triangle has the minimum perimeter of any triangle inscribed in a given acute triangle (Johnson 1929, pp. 161-165). The lengths of the legs of the orthic triangle are given by
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
The inradius of the orthic triangle is
 |
(6)
|
where 
is the circumradius of the reference triangle (Johnson 1929, p. 191), and the
circumradius is
 |
(7)
|
For an obtuse triangle or right triangle, the semiperimeter is
 |
(8)
|
which simplifies in the case of an acute triangle to
 |
(9)
|
where 
is the triangle area of 
and(Johnson 1929, p. 191).
Given a triangle 
,
construct the orthic triangle 
and determine the symmedian
points 
,
, and 
of 
, 
, and 
, respectively. Then the 
-symmedian of the corner triangle 
is the 
-median of triangle 
, and similarly for the (Honsberger 1995, p. 75).
In addition, the 
-median
of the corner triangle 
is the 
-symmedian
of triangle 
,
and similarly for the other two corner triangles.
Finally, the Euler lines of the three corner triangles 
, 
and 
pass through the Euler
points, and concur at a point 
on the nine-point circle
of triangle 
such that one of the following holds
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
(Thébault 1947, 1949; Thébault et al. 1951).
The sides of the orthic triangle are parallel to the tangents to the circumcircle at the vertices (Johnson 1929, p. 172). This is equivalent to the statement
that each line from a triangle's circumcenter to
a vertex is always perpendicular to the corresponding side of the orthic triangle
(Honsberger 1995, p. 22), and to the fact that the orthic and tangential
triangles are homothetic at Kimberling
center 
.
The triangle centroid of the orthic triangle has triangle center function
 |
(13)
|
(Casey 1893, Kimberling 1994), which is Kimberling center 
.
The symmedian point of the orthic triangle has
triangle center function
 |
(14)
|
(Casey 1893, Kimberling 1994), which is Kimberling center 
.
The following table gives the centers of the orthic triangle in terms of the centers of the reference triangle that correspond to
Kimberling centers 
.
 | center of orthic triangle |  | center of reference triangle |
 | incenter |  | radical center of (circumcircle, Parry circle, Bevan circle) |
 | triangle centroid |  | triangle centroid of orthic triangle |
 | circumcenter |  | nine-point center |
 | orthocenter |  | orthocenter of orthic triangle |
 | nine-point center |  | nine-point center of orthic triangle |
 | symmedian point |  | symmedian point of orthic triangle |
 | Euler infinity point |  | isogonal
conjugate of  |
 |  |  |  -of-orthic-triangle |
 | Tarry point |  |  -of-orthic-triangle |
 | Steiner point| |  |  -of-orthic-triangle |
 | psi(symmedian point, orthocenter) |  |  -of-orthic-triangle |
 | focus of Kiepert parabola |  |  -of-orthic-triangle |
 | Parry point |  |  -of-orthic-triangle |
 | psi(orthocenter, symmedian point) |  |  -of-orthic-triangle |
 | Parry reflection point |  | isogonal
conjugate of  |
 | isogonal
conjugate of  |  | Napoleon crossdifference |
