From Wikipedia, the free encyclopedia - View original article

"Orthocenter" and "Orthocentre" redirect here. For the orthocentric system, see Orthocentric system.

This article needs additional citations for verification. (January 2010) |

An altitude is the perpendicular segment from a vertex to its opposite side. In geometry, an **altitude** of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle 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 area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometric functions.

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. Also the altitude having the incongruent side as its base will form the angle bisector of the vertex.

It is common to mark the altitude with the letter *h* (as in *height*), often subscripted with the name of the side the altitude comes from.

In a right triangle, the altitude with the hypotenuse *c* as base divides the hypotenuse into two lengths *p* and *q*. If we denote the length of the altitude by *h*_{c}, we then have the relation

The three altitudes intersect in a single point, called the **orthocenter** of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. does not have an angle greater than or equal to a right angle). See also orthocentric system. If one angle is a right angle, the orthocenter coincides with the vertex of the right angle. Thus for acute and right triangles the feet of the altitudes all fall on the triangle's interior or edge.

The orthocenter, the centroid, the circumcenter and the 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.

The isogonal conjugate 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 system 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 sec *A* : sec *B* : sec *C* and barycentric coordinates

Denote the vertices of a triangle as *A*, *B*, and *C* and the orthocenter as *H*, and let *D*, *E*, and *F* denote the feet of the altitudes from *A*, *B*, and *C* respectively. Then:

- The sum of the ratios on the three altitudes of the distance of the orthocenter from the base to the length of the altitude is 1:
^{[1]}

- The sum of the ratios on the three altitudes of the distance of the orthocenter from the vertex to the length of the altitude is 2:
^{[1]}

- The product of the lengths of the segments that the orthocenter divides an altitude into is the same for all three altitudes:
^{[2]}

- If any altitude, say
*AD*, is extended to intersect the circumcircle at*P*, so that*AP*is a chord of the circumcircle, then the foot*D*bisects segment*HP*:^{[2]}

Denote the orthocenter of triangle *ABC* as *H*, denote the sidelengths as *a*, *b*, and *c*, and denote the circumradius of the triangle as *R*. Then^{[3]}^{[4]}

In addition, denoting *r* as the radius of the triangle's incircle, *r*_{a}, *r*_{b}, and *r*_{c} as the radii if its excircles, and *R* again as the radius of its circumcircle, the following relations hold regarding the distances of the orthocenter from the vertices:^{[5]}

If the triangle *ABC* is oblique (not right-angled), the points of intersection of the altitudes with the sides of the triangle 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 (that is, the center for the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.^{[6]}

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.

The orthic triangle provides the solution to Fagnano's problem, posed in 1775, of finding for the minimum perimeter triangle inscribed in a given acute-angle triangle.

The orthic triangle of an acute triangle gives a triangular light route.^{[7]}

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*

For any triangle with sides *a, b, c* and semiperimeter *s* = (*a+b+c*) / 2, the altitude from side *a* is given by

This follows from combining Heron's formula for the area of a triangle in terms of the sides with the area formula (1/2)×base×height, where the base is taken as side *a* and the height is the altitude from *a*.

Consider an arbitrary triangle with sides *a, b, c* and with corresponding altitudes *h*_{a}, *h*_{b}, and *h*_{c}. The altitudes and the incircle radius *r* are related by

Denoting the altitude from one side of a triangle as *h _{a}*, the other two sides as

Denoting the altitudes of any triangle from sides *a*, *b*, and *c* respectively as , , and ,and denoting the semi-sum of the reciprocals of the altitudes as we have^{[9]}

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. This is Viviani's theorem.

In a right triangle the three altitudes *h*_{a}, *h*_{b}, and *h*_{c} (the first two of which equal the leg lengths *b* and *a* respectively) are related according to^{[10]}^{[11]}

- ^
^{a}^{b}Panapoi,Ronnachai, "Some properties of the orthocenter of a triangle", University of Georgia. - ^
^{a}^{b}"Orthocenter of a triangle" **^**http://mathworld.wolfram.com/Orthocenter.html Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Web Resource.]**^**Altshiller-Court, Nathan,*College Geometry*, Dover Publications, 2007 (orig. Barnes & Noble 1952), p. 102.**^**Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization",*Forum Geometricorum*6, 2006, 335–342.**^**William H. Barker, Roger Howe (2007). "§ VI.2: The classical coincidences".*Continuous symmetry: from Euclid to Klein*. American Mathematical Society Bookstore. p. 292. ISBN 0-8218-3900-4. See also: Corollary 5.5, p. 318.**^**Bryant, V., and Bradley, H., "Triangular Light Routes,"*Mathematical Gazette*82, July 1998, 298-299.**^**Johnson, Roger A.,*Advanced Euclidean Geometry*, Dover Publ., 2007 (orig. 1929), p. 71.**^**Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle,"*Mathematical Gazette*89, November 2005, 494.**^**Voles, Roger, "Integer solutions of ,"*Mathematical Gazette*83, July 1999, 269–271.**^**Richinick, Jennifer, "The upside-down Pythagorean Theorem,"*Mathematical Gazette*92, July 2008, 313–317.

- 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, Wolfram Demonstrations Project.