In any acute triangle, the inscribed triangle with the smallest perimeter is the orthic triangle. The extended sides of the orthic triangle meet the opposite extended sides of its reference triangle at three collinear points. Trilinear coordinates for the vertices of the orthic triangle are given by Also, the incenter (the center of the inscribed circle) of the orthic triangle DEF is the orthocenter of the original triangle ABC. That is, the feet of the altitudes of an oblique triangle form the orthic triangle, DEF.

If the triangle ABC is oblique (does not contain a right-angle), the pedal triangle of the orthocenter of the original triangle is called the orthic triangle or altitude triangle. Triangle abc (respectively, DEF in the text) is the orthic triangle of triangle ABC Sec вЃЎ A : sec вЃЎ B : sec вЃЎ C = cos вЃЎ A в€’ sin вЃЎ B sin вЃЎ C : cos вЃЎ B в€’ sin вЃЎ C sin вЃЎ A : cos вЃЎ C в€’ sin вЃЎ A sin вЃЎ B, Orthic triangle The orthocenter has trilinear coordinates Let A, B, C denote the vertices and also the angles of the triangle, and let a = | BC|, b = | CA|, c = | AB| be the side lengths. If one angle is a right angle, the orthocenter coincides with the vertex at the right angle. does not have an angle greater than or equal to a right angle). The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. The three (possibly extended) altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H. Three altitudes intersecting at the orthocenter It is common to mark the altitude with the letter h (as in height), often subscripted with the name of the side the altitude is drawn to.

Also the altitude having the incongruent side as its base will be the angle bisector of the vertex angle. 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. The altitudes are also related to the sides of the triangle through the trigonometric functions. Thus, the longest altitude is perpendicular to the shortest side of the triangle. It is a special case of orthogonal projection.ДЄltitudes can be used in the computation of 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. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. The length of the altitude, often simply called 'the altitude', is the distance between the extended base and the vertex. The intersection of the extended base and the altitude is called the foot of the altitude. This line containing the opposite side is called the extended base of the altitude. In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle.

Median altitude geometry