**What’s going on:** Here we have a trapezoid and its diagonals. The segment parallel to the bases which passes through the intersection of the the diagonals is the harmonic mean of the lengths of the bases.

**Student handout: **HarmonicMeanInTrapezoid

**GeoGebra file:** here

## TEACHER NOTES

There are (at least) three sets of similar triangles. It is quite striking that HE and EI are congruent (making E a midpoint) with no congruent triangles in the figure. To prove this we want HI in terms of the length of the bases, a and b. This can be done by setting up proportions and combining them. When setting up the ratios both of the following may be useful (1) *top-portion-to-bottom-portion* ratios (which are a/b) and (2) *top-portion-to-‘whole-thing’* ratios (which are a/(a+b)).

**Important/Useful Takeaways for students:**

- The similar triangles give us helpful proportions. Then the power of algebra gives us many results.
- Proportions are quite easy to manipulate (with practice). There are useful properties of proportions.

The harmonic mean of two numbers is an interesting *mean. *The harmonic mean is *smaller* than the *arithmetic mean. *We will see that the harmonic mean is more ubiquitous than we might think.