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Altitude (triangle)
 
 
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In geometry
Geometry

Geometry 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 triangle is a straight line through a vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 and perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 to (i.e. forming a right angle
Right angle

In geometry and trigonometry, a right angle is an angle of 90 degree s, corresponding to a quarter turn . It can be defined; as the angle such that twice that angle amounts to a half turn, or 180?....
 with) the opposite side or an extension of the opposite side. The intersection between the (extended) side and the altitude is called the foot of the altitude.






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Triangle
In geometry
Geometry

Geometry 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 triangle is a straight line through a vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 and perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 to (i.e. forming a right angle
Right angle

In geometry and trigonometry, a right angle is an angle of 90 degree s, corresponding to a quarter turn . It can be defined; as the angle such that twice that angle amounts to a half turn, or 180?....
 with) the opposite side or an extension of the opposite side. The intersection between the (extended) side and the altitude is called the foot of the altitude. This opposite side is called the base of the altitude. The length of the altitude is the distance between the base and the vertex.

In an isosceles triangle (a triangle with two congruent sides), the altitude having the incongruent side as its base will have the midpoint of that side as its foot.

Altitudes can be used to compute the area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Through trigonometric functions, it can also give the length of one side of the triangle.

In a right triangle, 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 if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, 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 system
Orthocentric system

In geometry, an orthocentric system is a Set of four point in the plane one of which is the orthocenter of the triangle formed by the other three....
.

The orthocenter, along with the centroid
Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure is the intersection of all straight lines that divide into two parts of equal moment about the line....
, circumcenter and center of the nine-point circle
Nine-point circle

In 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, six lying on the triangle itself ....
 all lie on a single line, known as the Euler line. The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.

The isogonal conjugate
Isogonal conjugate

In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflection the lines PA, PB, and PC about the angle bisectors of A, B, and C....
 and also the complement
Complement (mathematics)

Complement has a variety of uses in mathematics:* complement, an operation that transforms an integer into its additive inverse, useful for subtracting numbers when only addition is possible, or is easier...
 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 system
Orthocentric system

In geometry, an orthocentric system is a Set of four point 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 coordinates
Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle....
 sec A : sec B : sec C and barycentric coordinates
Barycentric coordinates (mathematics)

In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates....




Orthic triangle


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 triangle
Pedal triangle

In 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....
 of the orthocenter of the original triangle. Also, the incenter 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 n LC, B" = LC n LA, C" = LC n LA. The tangential triangle, A"B"C", is homothetic to the orthic triangle.

The orthic triangle provides the solution to Fagnano
Giulio Carlo de' Toschi di Fagnano

Giulio 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.

Trilinear coordinates
Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle....
 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 coordinates
Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle....
 for the vertices of the tangential triangle are given by
  • A" = −a : b : c
  • B" = a : −b : c
  • C" = a : b : −c


Some additional altitude theorems


Equilateral triangle theorem:

For any point P within an equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
, 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 a, ß, ?. 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 hypotenuse
Hypotenuse

File:Triangle Sides.svgA hypotenuse is the longest side of a right 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 lengths of the other two sides....
, 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 a, ß, ?. The symphonic theorem states that triples (c2,h2,s2) and (a2,ß2,?2) are harmonic, and that triples and are Pythagorean:

External links

  • H. Lee Price and Frank R. Bernhart, Pythagorean triples and a new Pythagorean theorem, arxiv.org:math/0701554, (2007) ,[*symphonic theorem]
  • by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
  • With interactive animation
  • Compass and straightedge.
  • by Jay Warendorff, Wolfram Demonstrations Project
    Wolfram Demonstrations Project

    The Wolfram Demonstrations Project is a website developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience....
    .