As you may remember from my previous posts on this subject, I have been thinking a lot about the Common Core approach to secondary-school geometry, specifically the logical switcheroo that makes triangle congruence and similarity a consequence of assumptions about geometric transformations, rather than the other way around. To support this change, I have started to develop some curriculum, and I’m glad to say that it has been well received when I presented it to teachers in summer workshops.
I also wrote a document for teachers and curriculum developers, zeroing in on the logical issues implicit in the change. In writing it, I had two goals: choosing a limited set of logical assumptions for this new state of affairs, and presenting a pedagogically sound approach to get from those assumptions to the criteria for triangle congruence and similarity. My most significant contribution was to base the congruence criteria on a foundation of transformations, of course, but also geometric construction, which I believe puts this material within reach of a plausible classroom sequence.
To reach the similarity criteria, one needs some version of this result::
If O, A, and B are not collinear, the image A’B’ of the segment AB under a dilation with center O and scaling factor r is parallel to AB, with length r · AB.
In accord with the Common Core, I presented it as a foundational, unproved assumption. However, my colleague Lew Douglas has come up with an excellent proof of the result, which requires us to assume only that dilations preserve collinearity.
He and I collaborated on an update to my original paper, based on my approach to congruence and his approach to similarity. The new version is less polemical, and does not have as many links and footnotes, but it is logically stronger, more streamlined, and almost certainly a better guide to the topic for teachers and curriculum developers. Download it here.
(Of course, the nerdier among you may want to read both versions!)