Non-Coplanar "Quad" Midpoints?
- Tim Brzezinski
Creation of this applet was initially inspired by a Twitter conversation among Patrick Honner, Eddie Woo, Chris Bolognese, and Steven Strogatz. Here's the Twitter link. Before playing with the applet below, recall the theorem you've already discovered and proven in class and also illustrated here on this animation. (For a quick, informal investigation of this theorem, refer here.) Notice how on either worksheet, all 4 vertices of the original quadrilateral were coplanar. But what happens when we have 4 non-coplanar points? Check it out below: The red points shown are midpoints. Feel free to move the BIG WHITE VERTICES of the original "quadrilateral" anywhere you'd like!
To students familiar with 3-Dimensional Coordinate Geometry and working with vectors in 3-Space: How can you formally prove this theorem that holds true for the quadrilateral formed by the midpoints of consecutive segments formed by initially connecting any 4 non-coplanar points? That is, how can you formally prove what this applet informally illustrates?