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Unexpected applications of the fact that nth degree polynomimals are determined by n+1 points
I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof strategy works out, because I think such a collection of examples would be nice for impressing students of high school algebra.
I was trying to prove the "Power of a point" theorem (although I did not know it by that name at the time). My idea for a proof strategy was as follows:
After having this idea, we found the proof by similar triangles. While 3 values of $y$ are pretty easy to compute with, it was hard to find 9 easy ones.
So even though the method of proof didn't pan out in this case, it still seemed pretty cute, and hinted that "baby algebraic geometry" ideas could be presented to students in geometry or algebra classes this way.
I am asking for a list of similar situations where the proof strategy does pan out, so that they could be pulled out when teaching students of algebra or geometry