This conics investigation begins with a hands-on activity. Take a blank sheet of paper and mark a point F near the bottom of the paper that is roughly midway between the left and right edges.
As shown below, fold the paper so that a point on the bottom edge lands on point F. Make your crease sharp. Unfold the crease and continue to fold the rectangle so that new points on the bottom edge land on point F. What pattern do your creases seem to form?
The interactive picture below shows the Sketchpad way to model the Folded-Rectangle construction. To construct the crease formed when point A is folded onto point F, we draw segment AF and then construct its perpendicular bisector.
Drag point A and watch the crease pattern form. Now move point F to a new location. Clear the trace by clicking the red 'X' in the bottom right corner. Now drag point A again. Compare the new locus to the old one.
The distance definition of a parabola is: the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Assuming that the curve you've traced is a parabola, where do its focus and directrix appear to be?
The above model is fine, but it is rather annoying to redraw the locus each time point F's location changes. The interactive model below is an improvement. As you drag point F, all of the crease lines reposition themselves automatically.
It's time for a proof. How can you show that the shape outlined by the crease lines is a parabola? In the construction below, you'll see a line through point A perpendicular to the horizontal line. As you drag point A, notice that point B traces the curve in the picture. Point B is the tangent point on the crease.
Can you now prove that point B traces a parabola? Two hints: First, look for some congruent triangles. You'll need to add a segment to find them. Then, prove the curve is a parabola by showing that BF = BA.