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Altitude

In geometry, an altitude of a triangle is a straight line beginning at a vertex and ending 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 line is called the foot of the altitude. This side is called the base of the altitude. The length of the altitude is the distance between the base and the vertex.

Corollaries

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.

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 of a triangle intersect in a single point, called the orthocenter. 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).

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.