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Altitude (triangle)

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.

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

Altitudes can be used to compute the area of a triangle: it is given by one half of the product of length of altitude and its base.

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 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 bigger than a right one).

The orthocenter, along with the centroid, circumcenter and center of the nine point circle all lie on a single line, known as Euler's 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.

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.

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