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Trapezoid Divided into Equal Areas

by a Line Parallel to the Bases  -  The Root Mean Squared (RMS)

Given an isosceles trapezoid with parallel sides of length a and b.

A line of length c is drawn parallel to the bases of length a and b so as to divide the isosceles trapezoid into two equal areas.

Express c in terms of a and b.

See EMAT 4600/6600 problem Lines Parallel to the Bases of a Trapezoid.

This is the solution to the fourth part of that problem.  It corresponds to the Root Mean Squared (RMS)

Solution?

For a construction of the segment of length c, click here.

For GSP file of the completed construction, click here.

Extension.

Prove that for positive a and b,

with equality if and only if a = b.