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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. For a right triangle, the orthocenter coincides with the vertex at the right angle. [2]
The process of drawing the altitude from a vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection. Altitudes 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 (symbol b) equals the triangle's area: A = h b /2 ...
However, while the orthocenter and the circumcenter are in an acute triangle's interior, they are exterior to an obtuse triangle. The orthocenter is the intersection point of the triangle's three altitudes, each of which perpendicularly connects a side to the opposite vertex. In the case of an acute triangle, all three of these segments lie ...
This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by cyclic permutation of a, b, c. This process is known as cyclicity. [4] [5]
Note that the construction still works even if the orthocenter and circumcenter fall outside of the triangle. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
The three (possibly extended) altitudes intersect in a single point, called the orthocenter of the triangle. Fig 2. Orthocenters? It is a point all of three altitudes pass. In fact, existence of such a point is not proven beforehand. It might be like a definition of creature which has human face and lion body.
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.
M: circumcenter of ABC, orthocenter of DEF N: incenter of ABC, Nagel point of DEF S: centroid of ABC and DEF. The medial triangle can also be viewed as the image of triangle ABC transformed by a homothety centered at the centroid with ratio -1/2. Thus, the sides of the medial triangle are half and parallel to the corresponding sides of triangle ...