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The orthocenter of a triangle, usually denoted by H, is the point where the three (possibly extended) altitudes intersect. [ 1 ] [ 2 ] The orthocenter lies inside the triangle if and only if the triangle is acute .
Different functions may define the same triangle center. For example, the functions (,,) = and (,,) = both correspond to the centroid. Two triangle center functions define the same triangle center if and only if their ratio is a function symmetric in a, b, c.
The orthocenter of a triangle, usually denoted by H, is the point where the three (possibly extended) altitudes intersect. [1] [2] The orthocenter lies inside the triangle if and only if the triangle is acute. For a right triangle, the orthocenter coincides with the vertex at the right angle. [2]
Orthocentric system.Any point is the orthocenter of the triangle formed by the other three. In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three.
Sixteen key points of a triangle are its vertices, the midpoints of its sides, the feet of its altitudes, the feet of its internal angle bisectors, and its circumcenter, centroid, orthocenter, and incenter. These can be taken three at a time to yield 139 distinct nontrivial problems of constructing a triangle from three points. [12]
In geometry, the Euler line, named after Leonhard Euler (/ ˈ ɔɪ l ər / OY-lər), is a line determined from any triangle that is not equilateral.It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
where A, B, C denote both the triangle's vertices and the angle measures at those vertices; H is the orthocenter (the intersection of the triangle's altitudes); D, E, F are the feet of the altitudes from vertices A, B, C respectively; R is the triangle's circumradius (the radius of its circumscribed circle); and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C ...
The four altitudes of an orthogonal tetrahedron meet at its orthocenter. Edges AB, BC, CA are perpendicular to, respectively, edges CD, AD, BD. In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular.