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great so hopefully you're closer to developing a hypothesis for the relationship between the lengths of the segments in the diagram that describe exactly where the touching point is now recall that we're interested in the formula for a touching point because that'll let us write a computer program to compute just points on the parabolic arc and not have to worry about where the string aren't lines are or even where the control points are let me tell you a little bit about how I came up with my hypothesis so when we look at the diagram here with T set to the midpoint that is we're talking about a string art line that's halfway along the way in the construction that is this point is the midpoint of this segment this points the midpoint of this segment and finally the touching point looks like it's the midpoint of this string art line so in this case all those ratios are the same let's go back to say T equals around a quarter so this point is a quarter of along the way here this point is a quarter along the way here and in this case it looks like the touching point is also a quarter along the way of the string art line so again all the ratios are equal let's try one of their value of T say something around point seven so this point is point seven along the way this point is point seven along the way and again it looks like the touching point is about that same ratio about point seven along the way that leads us to guess that if this point is a fraction T along this line segment and this point is a fraction T along this line segment that the touching point that we're after is that same fraction along this string art line now to turn this into formulas let's start by labeling our points so let's call the control points a B and C this point here on a B I'll call point Q this point here on BC I'll call Point R and we'll call the touching point that we're after point P so the geometry says that if this is a fraction T along the way that this is in ratio T to one minus T this is in ratio T to one minus T and this is in ratio T to ist and the algebra that goes along with this geometry is that Q can be written as 1 minus T times pay plus T times B because it's on the line segment a B the point R is on the line segment BC a fraction T along the way so it can be written as 1 minus T times B plus T times C and if P is a fraction T along the line segment QR then it can be written again as 1 minus T times Q plus T times R so with these three formulas taken together we can compute any point on the parabola we like just by varying the value of T the next exercise will give you some practice using these formulas to solve a few problems