In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, an
altitude of a
triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
is a straight line through a
vertexIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
and
perpendicularIn geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to (i.e. forming a
right angleIn geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
with) a line containing the
base (the opposite side of the triangle). This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the
foot of the altitude. The length of the altitude, often simply called the altitude, is the distance between the base and the vertex. The process of drawing the altitude from the vertex to the foot is known as
dropping the altitude of that vertex. It is a special case of orthogonal projection.
Altitudes can be used to compute the
areaArea is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area, as well as being related to the sides of the triangle through trigonometric functions.
In an isosceles triangle (a triangle with two
congruentIn geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
sides), the altitude having the incongruent side as its base will have the
midpointThe midpoint is the middle point of a line segment. It is equidistant from both endpoints.-Formulas:...
of that side as its foot. Also the altitude having the incongruent side as its base will form the angle bisector of the vertex.
In a
right triangleA right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...
, the altitude with the hypotenuse as base divides the hypotenuse into two lengths
p and
q. If we denote the length of the altitude by
h, we then have the relation
- h2 = pq.
The orthocenter
The three altitudes intersect in a single point, called the
orthocenter of the triangle. The orthocenter lies inside the triangle (and consequently the feet of the altitudes all fall on the triangle)
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the triangle is not obtuse (i.e. does not have an angle greater than a right angle). See also
orthocentric systemIn geometry, an orthocentric system is a set of four points in the plane one of which is the orthocenter of the triangle formed by the other three....
.
The orthocenter, along with the
centroidIn geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
, circumcenter and center of the
nine-point circleIn geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points defined from the triangle...
all lie on a single line, known as the Euler line. The center of the nine-point circle lies at the
midpointThe midpoint is the middle point of a line segment. It is equidistant from both endpoints.-Formulas:...
between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
Unlike the
centroidIn geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
and circumcenter of a triangle, the orthocenter has no special characteristics (such as being equidistant from all sides or vertices).
The
isogonal conjugateIn geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P...
and also the complement of the orthocenter is the circumcenter.
Four points in the plane such that one of them is the orthocenter of the triangle formed by the other three are called an
orthocentric systemIn geometry, an orthocentric system is a set of four points in the plane one of which is the orthocenter of the triangle formed by the other three....
or orthocentric quadrangle.
Let
A,
B,
C denote the angles of the reference triangle, and let
a = |
BC|,
b = |
CA|,
c = |
AB| be the sidelengths. The orthocenter has
trilinear coordinatesIn geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...
sec
A : sec
B : sec
C and
barycentric coordinatesIn geometry, the barycentric coordinate system is a coordinate system in which the location of a point is specified as the center of mass, or barycenter, of masses placed at the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates...
-
Orthic triangle
If the triangle
ABC is oblique (not right-angled), the points of intersection of the altitudes with the sides of the triangles form another triangle, A'B'C', called the
orthic triangle or
altitude triangle. It is the
pedal triangleIn geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.More specifically, consider a triangle ABC, and a point P that is not one of the vertices A, B, C. Drop perpendiculars from P to the three sides of the triangle...
of the orthocenter of the original triangle. Also, the incenter (that is, the center for the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.
The orthic triangle is closely related to the
tangential triangle, constructed as follows: let
LA be the line tangent to the circumcircle of triangle
ABC at vertex
A, and define
LB and
LC analogously. Let
A" =
LB ∩
LC,
B" =
LC ∩
LA,
C" =
LC ∩
LA. The tangential triangle,
A"B"C", is homothetic to the orthic triangle.
The orthic triangle provides the solution to
FagnanoGiulio Carlo, Count Fagnano, and Marquis de Toschi was an Italian mathematician. He was probably the first to direct attention to the theory of elliptic integrals...
's problem which in 1775 asked for the minimum perimeter triangle inscribed in a given acute-angle triangle.
The orthic triangle of an acute triangle gives a triangular light route.
Trilinear coordinatesIn geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...
for the vertices of the orthic triangle are given by
- A' = 0 : sec B : sec C
- B' = sec A : 0 : sec C
- C' = sec A : sec B : 0
Trilinear coordinatesIn geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...
for the vertices of the tangential triangle are given by
- A" = −a : b : c
- B" = a : −b : c
- C" = a : b : −c
Equilateral triangle theorem
For any point P within an
equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.
Inradius theorems
Consider an arbitrary triangle with sides
a, b, c and with corresponding
altitudes
α, β, η. The altitudes and incircle radius
r are related by
-
-
Let
c, h, s be the sides of 3 squares associated with the right
triangle; the square on the
hypotenuseIn geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the...
, and the triangle's 2 inscribed
squares respectively. The sides of these squares
(c>h>s)
and the incircle radius
r are related by a similar formula:
-
-
The symphonic theorem
In the case of the right triangle, the sides of the 3 squares
c, h, s are
related to each other by the
symphonic theorem, as are the 3 altitudes
α, β, η. The
symphonic theorem states that triples
(c2,h2,s2) and
(α2,β2,η2) are
harmonic, and that triples
and
are
Pythagorean:
-
-
Area theorem
Denoting the altitudes from sides
a,
b, and
c respectively as
,
, and
,and denoting the semi-sum of the reciprocals of the altitudes as
we have
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