Rotating a parallelogram about its center
Introduction
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?