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congratulations you're now an expert on midpoints now let's take the next step I've been claiming that the curves created with the string our construction are parabolas but how do we know that's actually true well it's actually a little bit hard to prove that rigorously but we can get some intuition in the following exercise now you may have seen parabolic arcs as graphs of quadratic equations like this one for instance so here I have a green curve that's the graph of a quadratic function and so it's a parabola and I have a blue curve generated through the string art construction it's my job in this exercise it would be your job in the exercise to see if you can reposition the points so that the blue curve sits exactly on top of the green curve so for instance I'm gonna try moving this point say down here and this point over here over there so the blue curve is getting a little bit closer but it's not exactly on top of the green curve yet and in fact I'm going to ask you to try it in the next exercise