Editing Geometric mean (section)

===Geometry===
{{right_angle_altitude.svg}}
In the case of a [[right triangle]], its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude. This property is known as the [[geometric mean theorem]].

In an [[ellipse]], the [[semi-minor axis]] is the geometric mean of the maximum and minimum distances of the ellipse from a [[Focus (mathematics)|focus]]; it is also the geometric mean of the [[semi-major axis]] and the [[conic section#Conic parameters|semi-latus rectum]]. The [[semi-major axis]] of an ellipse is the geometric mean of the distance from the center to either focus and the distance from the center to either [[Directrix (conic section)|directrix]].
 
Another way to think about it is as follows:

Consider a circle with radius <math>r</math>. Now take two diametrically opposite points on the circle and apply pressure from both ends to deform it into an ellipse with semi-major and semi-minor axes of lengths <math>a</math> and <math>b</math>.
 
Since the area of the circle and the ellipse stays the same, we have:

: <math>
\begin{align}
\pi r^2 &= \pi a b \\

 r^2 &= a b \\
 
r &= \sqrt{a  b}
\end{align}
</math>

The radius of the circle is the geometric mean of the  semi-major and the semi-minor axes of the ellipse formed by deforming the circle.

Distance to the [[horizon]] of a [[sphere]] (ignoring the [[Horizon#Effect of atmospheric refraction|effect of atmospheric refraction]] when atmosphere is present) is equal to the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere.

The geometric mean is used in both in the approximation of [[squaring the circle]] by S.A. Ramanujan<ref>{{cite journal
 | last = Ramanujan | first = S. | author-link = Srinivasa Ramanujan
 | journal = [[Quarterly Journal of Mathematics]]
 | pages = 350–372
 | title = Modular equations and approximations to {{pi}}
 | url = http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf
 | volume = 45
 | year = 1914}}</ref> and in the construction of the [[Heptadecagon#Construction|heptadecagon]] with "mean proportionals".<ref>T.P. Stowell [https://books.google.com/books?id=qVfxAAAAMAAJ Extract from Leybourn's Math. Repository, 1818] in ''The Analyst'' via [[Google Books]]</ref>