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In geometry, an altitude of a triangle is a line segment through a given vertex (called apex) and perpendicular to a line containing the side or edge opposite the apex. This (finite) edge and (infinite) line extension are called, respectively, the base and extended base of the altitude.
Any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle. The theorem can also be thought of as a special case of the intersecting chords theorem for a circle, since the converse of Thales' theorem ensures that the hypotenuse of the right angled triangle is the diameter of its ...
The Simson line of a vertex of the triangle is the altitude of the triangle dropped from that vertex, and the Simson line of the point diametrically opposite to the vertex is the side of the triangle opposite to that vertex. If P and Q are points on the circumcircle, then the angle between the Simson lines of P and Q is half the angle of the ...
That is, the feet of the altitudes of an oblique triangle form the orthic triangle, DEF. Also, the incenter (the center of the inscribed circle) of the orthic triangle DEF is the orthocenter of the original triangle ABC. [22] Trilinear coordinates for the vertices of the orthic triangle are given by =: : = :: = : :
The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's medial triangle (with vertices at the midpoints of the sides of the reference triangle) and its orthic triangle (with vertices at the feet of the reference triangle's altitudes). [6]: p.153
The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180 ...
They went from being wild animals to becoming man’s best friend. And some people even believe we don’t actually deserve them. Dogs have developed a well-deserved reputation as being loyal ...
The altitude to the hypotenuse is the geometric mean (mean proportional) of the two segments of the hypotenuse. [2]: 243 Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. In equations, =, (this is sometimes known as the right triangle altitude theorem)
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