Instead of working through the problems in the geometry book, we decided it would be fun to try to prove all the theorems as we come to them. A couple of weeks ago he proved the midsegment theorem. Now we are on to trying to prove that the centroid is 2/3 of the way down the median. So we sit around the white board easel and discuss how we might go about this. At this point, I don't even know how we're gonna prove this thing. We learned from the midsegment theorem that defining the vertices using (x1, y1), (x2, y2), (x3, y3) helped out greatly. So, what the heck, let's try it again.
We know we can define the midpoints generally as well, so that's what we do. Then it hits! We can define the vector from the vertex to the midtpoint of the opposite side, use a scalar of 2/3 to determine the vector from the vertex to what's supposed to be the centroid and then translate the vertex to said point.
I didn't know that. If this is what it's like to become a co-learner with your students, then sign me the heck up.