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In Euclidean geometry, the right triangle altitude theorem or geometric mean theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of those two segments equals the altitude.
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.
The exact definition and reference datum varies according to the context (e.g., aviation, geometry, geographical survey, sport, or atmospheric pressure). Although the term altitude is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage.
If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other. From this: The altitude to the hypotenuse is the geometric mean (mean proportional) of the two segments of the hypotenuse. [2]: 243
Height is also used as a name for some more abstract definitions. These include: The height or altitude of a triangle, which is the length from a vertex of a triangle to the line formed by the opposite side; The height of a pyramid, which is the smallest distance from the apex to the base;
The geometric mean of the three numbers is the cube root of their product, for example with numbers , , and , the geometric mean is = =. The geometric mean is useful whenever the quantities to be averaged combine multiplicatively, such as population growth rates or interest rates of a financial investment.
altitude: "height where the chosen reference surface is mean sea level" [1] The International Civil Aviation Organization (ICAO) offers similar definitions: [2] altitude: "the vertical distance of a level, a point or an object considered as a point, measured from the mean sea level (MSL);" [2]
For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line.
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