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# 4. How can we prove this?

## Video transcript

well you should now have some experience in computing points on parabolas using the formulas that we wrote down in the previous video these formulas are based on the hypothesis that all of these ratios are in the same proportion and that proportion is governed by T but now we're going to prove that these formulas are actually correct so I'm going to use this version of the interactive in the proof you'll have a chance to experiment with this interactive in a minute and as before I've got a string art line it is controlled by a parameter T as I wiggle T back and forth the string aren't lying Wiggles back and forth the method I'm going to use to find where the touching point occurs seems a little bit sneaky at first but it's the simplest method I know and what I'm going to do is introduce another string art line this one controlled by a parameter s and what I'm going to do is I'm going to write down an expression for this intersection point here this Green Point now why would I do that well the reason is watch what happens is s and T get closer and closer together so as I make s closer and closer to t watch what happens to this intersection point okay it moves and it gets closer closer to the parabola and exactly when s equals T that intersection point lies exactly on the parabola so if I can write down a formula for where that intersection point is I can compute exactly where the touching point occurs this is probably a good time to pause and let you experiment with this interactive to get some feeling for what we're about to do algebraically