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Dividing a Trapezium

Students may learn the formula for finding the area of a trapezium.
This applet demonstrate that the simple area formula can be applied
to solve this not so simple problem.

http://www.geogebratube.org/material/show/id/15772 (Download GeoGebra file)

It is possible to divide the area of a trapezium into two equal halves
by a straight line.

You may move point E in the applet and feel it.
Try to figure out your own method and explain.

* In North America, trapezium is known as trapezoid *

If other conditions are given for this area dividing problem, it will become more interesting (or challenging).

For example, how to divide the area of the trapezium into two equal halves with a line parallel to AB?