Altitude (triangle) - The Orthocenter

The Orthocenter

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.

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. 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:

• 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:

• The product of the lengths of the segments that the orthocenter divides an altitude into is the same for all three altitudes:

• 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:

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

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: