Illustrating Thales' Theorem and the Varignon Quadrilateral

Thales' Theorem: In a circle, any triangle involving the endpoints of the diameter and a point on a circle will be a right angle.

The Varignon Parallelogram: In a convex quadrilateral, the midpoints of the side create a parallelogram.

Manipulate B, D, and E as you look at the questions below the mathlet.

1) Notice what the angle measures at D and E on the circle remain no matter how you move those two points.

2) Notice that the areas of the two colored triangles, despite having a common side, do not usually have common areas. Why? The heights of the triangles can differ. If the triangles are not congruent, Quadrilateral BDCE can be irregular.

3) Points F, G, H, I are all midpoints of Quadrilateral BDCE. Notice the interior angle measures of Quadrilateral FGHI, and notice what type of special quadrilateral it stays no matter how you move B and D.

Darron Steele, Created with GeoGebra