this was my comment on Dan Burfeind's blog. It has been too long since I copied it back here for me to remember all the editing intentions I had, so, I simply offer as-is:

For another great coordinate geometry investigation, you could spring off this numberphile video: All Triangles are Equilateral, which I saw in a post from Mike Lawler. Can the kids use coordinate geometry to investigate where the proof goes wrong? I’d suggest they follow Mike’s son and use a right triangle (even 3-4-5 as in the blog video).

Choosing a right triangle and setting it in the coordinate plane carefully (right angle at the origin, legs along the axes) makes the equations really nice and, I think, forms a useful demonstration of the power of coordinate geometry. If they play with the picture, there are further rewards as they see interesting relationships between triangle side lengths and the lengths of other key line segments.

A further tidbit out of this whole sequence is that you can also have an interesting discussion about what it means for something to be a proof. The numberphile argument looks, smells, and feels like a solid proof, but we know it is wrong. Something subtle was left out, but there are always subtle things left out of real proofs. Furthermore, we start with the feeling that, where two lines intersect is minor, technical and obvious. Then, it is suddenly a lot more difficult when we try to prove it. Then it turns out the original idea was wrong.

A classic progression: “obvious, hard, false.”

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