The Harmonic Mean

A very wordy definition of the Harmonic Mean of two positive numbers  a  and  b where  a  and  b are the smallest and largest, respectively, of three numbers and the third is the HM.

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




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).  See also Parallel Line Segment Through the Intersection of Diagonals of a Trapezoid




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. This was examined int the Combining Rates problem. The elaboration of average rate yielding the harmonic mean is found in that problem of travel from Athens to Danielsville and return.