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Hard, but Achievable
Sep. 26th, 2017 08:41 amWe had a bit of fun in yesterday's geometry class. We were finishing up the section on triangle geometry, with the capstone being the proof of the existence of the Nine Point Circle. That done, I began showing them some odds and ends about the various points we'd been discussing - the circumcenter O, the orthocenter H, and the nine-point center N. Using Sketchpad, I'd set up a dynamic sketch displaying all of those points, and how they behaved as the triangle ABC changed shape.
One of the students then asked if it was possible for the vertex A to coincide with N. I said, "I don't know. Let's find out!". I grabbed A with the mouse and pulled it towards N. The latter point also moved, of course, but more slowly, and I was able to catch up. I couldn't get them to coincide exactly, but that was a matter of human clumsiness - clearly they could be made to match. I had a doodad off to the side keeping track of the angle BAC; one of the students pointed out that the angle was almost exactly 120 degrees. 120 degrees being a Significant Angle, I decided this couldn't be a coincidence. Then, the student who had suggested the problem pointed out that the triangle was demonstrably (not just apparently) isosceles, and we saw that that meant the triangle HBC was equilateral! I realized that this tied in to another interesting set of ideas, which I pointed out as time ran out.
Understanding may not be teachable, but it is learnable!
One of the students then asked if it was possible for the vertex A to coincide with N. I said, "I don't know. Let's find out!". I grabbed A with the mouse and pulled it towards N. The latter point also moved, of course, but more slowly, and I was able to catch up. I couldn't get them to coincide exactly, but that was a matter of human clumsiness - clearly they could be made to match. I had a doodad off to the side keeping track of the angle BAC; one of the students pointed out that the angle was almost exactly 120 degrees. 120 degrees being a Significant Angle, I decided this couldn't be a coincidence. Then, the student who had suggested the problem pointed out that the triangle was demonstrably (not just apparently) isosceles, and we saw that that meant the triangle HBC was equilateral! I realized that this tied in to another interesting set of ideas, which I pointed out as time ran out.
Understanding may not be teachable, but it is learnable!
