Editing Harmonic mean (section)

===In geometry===
In any [[triangle]], the radius of the [[incircle and excircles of a triangle|incircle]] is one-third of the harmonic mean of the [[Altitude (triangle)|altitudes]].

For any point P on the [[Arc (geometry)|minor arc]] BC of the [[circumcircle]] of an [[equilateral triangle]] ABC, with distances ''q'' and ''t'' from B and C respectively, and with the intersection of PA and BC being at a distance ''y'' from point P, we have that ''y'' is half the harmonic mean of ''q'' and ''t''.<ref>{{cite book |last1=Posamentier |first1=Alfred S. |last2=Salkind |first2=Charles T. |title=Challenging Problems in Geometry |edition=Second |publisher=Dover |year=1996 |page=[https://archive.org/details/challengingprobl0000posa/page/172 172] |isbn=0-486-69154-3 |url=https://archive.org/details/challengingprobl0000posa |url-access=registration }}</ref>

In a [[right triangle]] with legs ''a'' and ''b'' and [[Altitude (triangle)|altitude]] ''h'' from the [[hypotenuse]] to the right angle, {{math|''h''<sup>2</sup>}} is half the harmonic mean of {{math|''a''<sup>2</sup>}} and {{math|''b''<sup>2</sup>}}.<ref>Voles, Roger, "Integer solutions of <math>a^{-2}+b^{-2}=d^{-2}</math>," ''Mathematical Gazette'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–;317.</ref>

Let ''t'' and ''s'' (''t'' > ''s'') be the sides of the two [[Triangle#Squares|inscribed squares in a right triangle]] with hypotenuse ''c''. Then {{math|''s''<sup>2</sup>}} equals half the harmonic mean of {{math|''c''<sup>2</sup>}} and {{math|''t''<sup>2</sup>}}.

Let a [[trapezoid]] have vertices A, B, C, and D in sequence and have parallel sides AB and CD. Let E be the intersection of the [[diagonal]]s, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC. (This is provable using similar triangles.)

[[File:CrossedLadders.png|thumb|right|Crossed ladders. ''h'' is half the harmonic mean of ''A'' and ''B'']]
One application of this trapezoid result is in the [[crossed ladders problem]], where two ladders lie oppositely across an alley, each with feet at the base of one sidewall, with one leaning against a wall at height ''A'' and the other leaning against the opposite wall at height ''B'', as shown. The ladders cross at a height of ''h'' above the alley floor. Then ''h'' is half the harmonic mean of ''A'' and ''B''. This result still holds if the walls are slanted but still parallel and the "heights" ''A'', ''B'', and ''h'' are measured as distances from the floor along lines parallel to the walls. This can be proved easily using the area formula of a trapezoid and area addition formula.

In an [[ellipse]], the [[semi-latus rectum]] (the distance from a focus to the ellipse along a line parallel to the minor axis) is the harmonic mean of the maximum and minimum distances of the ellipse from a focus.