The Harmonic Mean

 If three numbers are such that by whatever part of itself the largest term exceeds the middle term, and the middle term exceeds the third by the same part of the third then the middle term is the harmonic mean of the first and third. This is the relationship shown in this equation where b is the largest term, c is the middle term and a is the smallest term.

Usually the relation is written in one of the following forms to show that c is the harmonic mean of positive numbers a and b:

or

 

Thus the length of the parallel line segment through the intersection of the diagonals is the harmonic mean of the bases of the trapezoid. Click here for a demonstration (use the Back key to retrun to this point).

 

The harmonic mean is also used to find the average rate. For example, if the rate for one lap at the race trace is a and the rate for a second lap is b then the average rate c is given by the harmonic mean. See average rate.

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