In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle 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. Image File history File links Triangle. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ... Fig. ... This article is about angles in geometry. ...

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. See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...

Altitudes can be used to compute the area 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. Area is a physical quantity expressing the size of a part of a surface. ...

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 For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

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 the triangle is not obtuse (i.e. does not have an angle greater than a right angle). See also orthocentric system. It has been suggested that this article or section be merged with Logical biconditional. ... In geometry, an orthocentric system is a set of four points in the plane where one point is the orthocenter of the triangle formed by the other three. ...

The orthocenter, along with the centroid, circumcenter and center of the nine-point circle 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. Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... In geometry, the nine-point circle is a circle that can be constructed for any given triangle. ... In geometry, Eulers line (red line in the image), named after Leonhard Euler, is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine-point circle (red point) of any triangle. ...

The isogonal conjugate of the orthocenter is the circumcenter. In 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. The isogonal conjugate of a... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...

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 or orthocentric quadrangle. In geometry, an orthocentric system is a set of four points in the plane where one point is the orthocenter of the triangle formed by the other three. ...

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 sec A : sec B : sec C and barycentric coordinates Trilinear coordinates describe the relative distances from the three sides of a given triangle. ... In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex. ...

((a² + b² − c²)(a² − b² + c²):(a² + b² − c²)( − a² + b² + c²):(a² − b² + c²)( − a² + b² + c²)).

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 of the orthocenter of the original triangle. Also, the incenter of the orthic triangle is the orthocenter of the original triangle. In geometry, given a triangle and a point, the pedal triangle is given thus: Let the triangle be ABC, and the point P. Drop perpendiculars from P to the three sides of the triangle (these may need to be produced, ie extended). ...

The orthic triangle is closely related to the tangential triangle, constructed as follows: let L_A be the line tangent to the circumcircle of triangle ABC at vertex A, and define L_B and L_C analogously. Let A" = L_B∩L_C, B" = L_C∩L_A, C" = L_C∩L_A. The tangential triangle, A"B"C", is homothetic to the orthic triangle. In economics, utility is a measure of the happiness or satisfaction gained from a good or service. ...

The orthic triangle provides the solution to Fagnano's problem which in 1775 asked for the minimum perimeter triangle inscribed in a given acute-angle triangle. Giulio Carlo, Count Fagnano, and Marquis de Toschi, (December 6, 1682 Sinigaglia - September 26, 1766) was an Italian mathematician. ...

Trilinear coordinates for the vertices of the orthic triangle are given by Trilinear coordinates describe the relative distances from the three sides of a given triangle. ...

• A' = 0 : sec B : sec C
• B' = sec A : 0 : sec C
• C' = sec A : sec B : 0

Trilinear coordinates for the vertices of the tangential triangle are given by Trilinear coordinates describe the relative distances from the three sides of a given triangle. ...

• 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, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

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: It has been suggested that this article or section be merged into Stanford University. ...

Let c, h, s be the sides of 3 squares associated with the right triangle; the square on the hypotenuse, 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: A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ...

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:

External links

• H. Lee Price and Frank R. Bernhart, Pythagorean triples and a new Pythagorean theorem, arxiv.org:math/0701554, (2007) [1],[*symphonic theorem]
• Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Orthocenter of a triangle With interactive animation
• Animated demonstration of orthocenter construction Compass and straightedge.