[color=#000000]Creation of this applet was initially inspired by a Twitter conversation among [url=https://twitter.com/MrHonner]Patrick Honner[/url], [url=https://twitter.com/misterwootube]Eddie Woo[/url], [url=https://twitter.com/EulersNephew]Chris Bolognese[/url], and [url=https://twitter.com/stevenstrogatz]Steven Strogatz[/url]. Here's the [url=https://twitter.com/MrHonner/status/762796821756387328]Twitter link[/url]. [br][br][br]Before playing with the applet below, recall the theorem you've already discovered and proven in class and also illustrated here on [url=https://www.geogebra.org/m/uNW647XY]this animation[/url]. (For a quick, informal investigation of this theorem, refer [url=https://www.geogebra.org/m/xGxYdjWX]here[/url].) [br]Notice how on either worksheet, all 4 vertices of the original quadrilateral were coplanar. [br][/color][i][color=#0000ff][b]But what happens when we have 4 non-coplanar points?[/b][/color][/i][color=#000000] Check it out below: [br][/color][color=#000000][br]The [/color][b][color=#cc0000]red points[/color][/b][color=#000000] shown are [/color][color=#cc0000][b]midpoints.[/b][/color][color=#000000] [br]Feel free to move the [b]BIG WHITE VERTICES[/b] of the original "quadrilateral" [b]anywhere you'd like! [/b] [br][/color]

[color=#000000]To students familiar with 3-Dimensional Coordinate Geometry and working with vectors in 3-Space: [br][br]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? [/color]