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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.
PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ. Visual proof that (x + y) 2 ≥ 4xy. Taking square roots and dividing by two gives the AM–GM inequality. [1]
Suppose AC = x 1 and BC = x 2. Construct perpendiculars to [AB] at D and C respectively. Join [CE] and [DF] and further construct a perpendicular [CG] to [DF] at G. Then the length of GF can be calculated to be the harmonic mean, CF to be the geometric mean, DE to be the arithmetic mean, and CE to be the quadratic mean.
Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a : b , AO ≥ GQ. Geometric proof without words that max ( a , b ) > root mean square ( RMS ) or quadratic mean ( QM ) > arithmetic mean ( AM ) > geometric mean ( GM ) > harmonic mean ( HM ) > min ( a , b ) of two distinct positive numbers a and b ...
In mathematics, the arithmetic–geometric mean (AGM or agM [1]) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential , trigonometric functions , and other special functions , as well as some ...
The power mean could be generalized further to the generalized f-mean: (, …,) = (= ()) This covers the geometric mean without using a limit with f(x) = log(x). The power mean is obtained for f(x) = x p. Properties of these means are studied in de Carvalho (2016).
Jordan–Schönflies theorem (geometric topology) JSJ theorem (3-manifolds) Lickorish twist theorem (geometric topology) Lickorish–Wallace theorem (3-manifolds) Nielsen realization problem (geometric topology) Nielsen-Thurston classification (low-dimensional topology) Novikov's compact leaf theorem ; Perelman's Geometrization theorem (3 ...
Balancing: An interesting problem is whether this condition (together with symmetry, fixed-point, monotonicity and continuity properties) implies that the mean is quasi-arithmetic. Georg Aumann showed in the 1930s that the answer is no in general, [ 7 ] but that if one additionally assumes M {\displaystyle M} to be an analytic function then the ...