In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, a vertex (plural "vertices" is a special kind of point. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent e. forming a right angle with) the opposite side or an extension of the opposite side. In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle 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 of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's 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. 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. Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised If we denote the length of the altitude by h, we then have the relation
- h2 = pq.
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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. In Geometry, an orthocentric system is a set of four points in the plane one of which is the Orthocenter of the triangle
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. In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon In Geometry, the nine-point circle is a Circle that can be constructed for any given triangle. In Geometry, the Euler line, named after Leonhard Euler, is a line determined from any Triangle that is not equilateral; it passes 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 and also the complement 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 In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon
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 one of which is the Orthocenter of the triangle
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
- ((a2 + b2 − c2)(a2 − b2 + c2):(a2 + b2 − c2)( − a2 + b2 + c2):(a2 − b2 + c2)( − a2 + b2 + c2)). In Geometry, the trilinear coordinates of a point relative to a given Triangle describe the relative distances from the three sides of the triangle In Mathematics, barycentric coordinates are coordinates defined by the vertices of a Simplex (a triangle Tetrahedron, etc
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. In Geometry, a pedal triangle is obtained by projecting a point onto the sides of a 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 ∩ LC, B" = LC ∩ LA, C" = LC ∩ LA. The tangential triangle, A"B"C", is homothetic to the orthic triangle. In Mathematics, a homothety (or homothecy or dilation) is a transformation of space which takes each line into a parallel line (in essence a
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 - September 26, 1766) was an Italian Mathematician
Trilinear 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 coordinates 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, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. In Geometry, the trilinear coordinates of a point relative to a given Triangle describe the relative distances from the three sides of the triangle In Geometry, the trilinear coordinates of a point relative to a given Triangle describe the relative distances from the three sides of the triangle Properties The area of an equilateral triangle with sides of length a\\!
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 hypotenuse, and the triangle's 2 inscribed squares respectively. In Geometry, the incircle or inscribed circle of a triangle is the largest Circle contained in the triangle it touches (is Tangent A hypotenuse is the longest side of a Right triangle, the side opposite of the Right angle. 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:
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.
- An interactive Java applet for the orthocenter
- Fagnano's Problem by Jay Warendorff, The Wolfram Demonstrations Project.
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