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Rotating a parallelogram about its center


Here is parallelogram ABCD. It was constructed with free points A, B, and C. The fourth vertex, D, is defined as A-B+C. (This is the fastest GeoGebra parallelogram construction I know.) The slider controls a rotation that produces parallelogram A'B'C'D'. The center of rotation, E, is defined as the midpoint of AC. So far as we know, it's just a coincidence that it's also on BD.
Suppose we know that 1) A half-turn reverses the vector from the center of rotation to any point; 2) A rotation is an isometry (that is, it preserves all lengths); 2) The midpoint of two points is their average. Let be the half-turn about E; so, for example, .

How can we show that E is the midpoint of BD?

How can we show that sends A to C?

Similarly, sends B to D.

How can we show that AB=CD?

Similarly: AE=CE; BE=DE; AD=BC.

In general, any object is congruent to its image under an isometry; that includes polygons, angles, etc. What pairs of objects are congruent under this isometry?