Yesterday (11/5/2014) Numberphile published a really neat “proof” that all triangles are equilateral:

My older son and I are in the middle of reviewing the chapter in our geometry book about similar triangles, so this new video from Numberphile was well-timed for me. We watched the video last night and today I used it as a way to review a little bit about similar triangles and constructions.

As usual, all of this was done on the fly and is pretty raw. One particular bit of unluckiness was my choice a 3-4-5 triangle for this exercise, but since the main point was review anyway I’m not dwelling on that poor choice too much. Should you want to do a similar exercise, though, definitely choose a different triangle 🙂

Despite how raw this project is, it is a pretty good struggle. After we finished my son said “that was hard.” He’s right, but that was part of the point. The Numberphile video flows really well but that great flow hides a lot of geometry that is not super easy to work through – especially for someone just learning the subject. What makes this project especially fun and hopefully useful is that this math is accessible to a geometry student with just a little bit of work.

So, with the warning about the rawness out of the way, we began by talking about the problem and attempting the first construction – construct a (non-equilateral) triangle given three sides:

Next we talked about the construction of the perpendicular bisector of the long side of the triangle. My son was able to remember how that construction worked which made this part go much more smoothly than the later parts of the exercise. After the construction, though, I asked how he knew that the perpendicular bisector was an equal distance from two nearby vertices of the triangle. That geometric insight was tough for him. Tough enough, in fact, that I paused the recording to give him some more time to think about it.

Returning from the pause he is able to see the congruent triangles that make the lengths from the triangles vertices equal. I’m glad that he was able to make it through this little struggle and eventually see these congruent triangles. Until walking through this geometry book with my son I’d never really thought about how connected similar triangles were with constructions.

Following the discussion about the perpendicular bisector, we drew in the angle bisector using a protractor. The reason I used a protractor here is that I did not think we would have enough time to include the angle bisector construction in the 45 minute time period we had this morning. Having completed the project and seen how difficult it was, I’m not sure that I’d make a different choice now if I had more time.

In an unlucky coincidence the angle bisector we draw intersects the perpendicular bisector at one of the points we’d drawn in previously. You can see in the Numberphile video that they had faintly drawn in all of the lines they needed ahead of time. You can see in my video what happens when you don’t do that 🙂

The next part is the critical piece in finding the flaw in the Numberphile “proof,” and it turns out to be the most difficult piece for my son to understand. The math problem here is figuring out how to construct a perpendicular line segment from a point to a given line. The construction itself is pretty similar to the construction of the perpendicular bisector of a line segment, but if you’ve not tried out this construction before this similarity is far from obvious. In this video we discuss the problem but do not solve it.

We talked about how to do the construction for probably 10 minutes after I turned off the camera.

After our 10 minute discussion he was ready to try out the construction on his own. Even with the long discussion leading up to this part of the project, the ideas are not totally clear in his mind. It is nice to see him work them out on the fly, though.

More than anything else, I think the lesson here is that what seems like a really simple step in the Numberphile proof , just dropping these perpendicular lines to the (extended) sides of the original triangle, has some difficult geometric ideas behind it. Working through these ideas takes some time – much much more time than I expected, in fact – but hopefully leads to a better understanding of geometry and constructions.

Now we return to our original diagram to construct the perpendicular lines we’ve spent the last 15 minutes discussing. As with the practice construction in the last video this one isn’t without a little difficulty. But we get there. Also, we do get a little unlucky with our 3-4-5 triangle and one of the perpendiculars doesn’t quite come from the construction we’d been doing.

The final step now (and this is the final step just because we were coming up on our 45 min time limit) is to notice the difference between our picture and the picture at this step in the Numberphile video. He doesn’t see the difference at first, so we pause and re watch the relevant part of the Numberphile video:

Now we return and he notices that the perpendiculars are not in the same place that they were in the video. Something is amiss. I’m not super happy with how I explain the flaw in the proof at the end, but that’s fine. The main point was working through the constructions rather than understanding the flaw. I’m pretty sure that he didn’t believe that all triangles were equilateral anyway!

So a fun and challenging project. With a bit more prep I think this could have been quite a bit better. Even the relatively raw approach shows how useful this Numberphile video could be to kids learning geometry, though. At least I hope so!

## Comments

Two points on the theme of angular bisectrix.

A) Construction. Leaving that for the dad, of you two, but you need to use the compass three times and the ruler once.

B) Perpendicular … perpendicular to surrounding lines onto it – or perpendicular to it and onto surrounding sides. Which is easier to construct onto a specific point?

Numberphile used the first, but either way you will get equal distances to the surrounding lines.

For my own part, I also got a C* between A and C, and I also used a 3 – 4 – 5, but I had AB as three, BC as four and therefore bisectrix at M parallel to prolongation of AB.

You might want to revisit this set of constructions to investigate whether the perpendicular really has its foot at the vertex of the original triangle. Some ideas:

(1) think of a point P on the angle bisector with perpendiculars to the two rays of the angle and line segments to the other two vertices of the triangle. What happens to the picture if you slide P out along the bisector?

(2) draw the picture using an online construction tool

(3) use coordinate geometry. If you set up the right angle at the origin and legs along the axes then all the equations are really nice.

Along the way, you should see an interesting relationship about where the feet of the perpendiculars actually are. Then, reverse the question and ask what shape right triangle will actually have one of those feet land on the triangle vertex.

In case you haven’t already seen it, take a look at Euclid: The Game (a free geogebra-based game of classical constructions).

Interesting idea. My son was just starting to look at constructions when we did this project. Today we are in the review section in the chapter about triangles, and he’s got quite a bit more experience with constructions. Might be fun to go back and revisit this project now.

At the time several teachers told me that they thought exploring the video via the online construction tools would be a much better way to go, so you aren’t alone in recommending #2!

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