Creation of this applet was inspired by a [url=https://twitter.com/gogeometry/status/198099723184377857]tweet[/url] from [url=https://twitter.com/gogeometry]Antonio Gutierrez[/url] (GoGeometry). [br][br]In addition to proving that the [b][color=#ff00ff]pink point is the midpoint[/color][/b] of [color=#38761d][b]this segment[/b][/color] [b][color=#1e84cc]drawn parallel to both bases[/color][/b], [color=#38761d][b]prove that this segment has a length equal to the harmonic mean[/b][/color] of the trapezoid's base lengths. [br][br]Recall the [b][color=#38761d]harmonic mean of 2 numbers [i]a[/i] and [i]b[/i][/color][/b] is defined to be the [b][color=#38761d]reciprocal of the average (arithmetic mean) of these numbers' reciprocals. [/color][/b][br][br]In essence, the [b][color=#38761d]harmonic mean of [i]a[/i] and [i]b[/i][/color][/b] = [math]\frac{1}{\frac{\left(\frac{1}{a}+\frac{1}{b}\right)}{2}}[/math] = [math]\frac{2ab}{a+b}[/math].