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Standing on the Shoulders of Giants

Here is a line and a point C not on the line. Use straightedge and compass tools to construct a line perpendicular to line that goes through point C.   Submit your reasoning for the actions taken and try to justify your results.
Are you ready for more?
  1. The line segment AB has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment AB. How far is this new point from A?
  2. We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment AB. How far is this new point from A?
  3. If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?