The[b][color=#9900ff] harmonic mean of 2 numbers [i]a[/i] and [i]b[/i][/color][/b] is defined to be the [b][color=#9900ff]reciprocal of the average (arithmetic mean) of these numbers' reciprocals. [/color][/b][br][br]In essence, the [b][color=#9900ff]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]. [br][br][br]Prove that [b][color=#9900ff]the segment drawn through the point of intersection[/color][/b] [color=#1e84cc][b]of both diagonals[/b][/color] [color=#38761d][b]of an isosceles trapezoid[/b][/color] [b][color=#9900ff]that is parallel to both bases[/color][/b] is equal to the [b][color=#9900ff]harmonic mean[/color][/b] [b][color=#666666]of the bases [/color][color=#38761d]of this isosceles trapezoid. [br][br][/color]Can you prove this is true for ANY TRAPEZOID (and not just an isosceles trapezoid?) [/b]