This conics investigation begins with a hands-on activity. Using a piece of paper, draw yourself a circle with at least a 3-inch radius and cut it out. Mark the center point C clearly. Now pick a random point on the circle and mark it as point F.
As shown below, fold the circle so that a point on the circle's circumference lands on point F. Mark your crease sharply. Unfold the crease and continue to fold and unfold the circle so that new points on the circle's circumference land on point F. What pattern do your creases seem to form?
The interactive model below shows the Sketchpad way to model the Folded-Circle construction. In order to construct the crease formed when point A is folded onto point F, we draw segment AF and then construct its perpendicular bisector.
As you drag point A around the circle, you'll see the crease pattern form. Now move point F to a new location. To clear the trace, click the red 'X' in the bottom right corner. Drag point A again. Compare the new locus to the old one.
Describe how the locus varies depending on point F's location. What happens when F is very close to C? What happens when F is near the circle's circumference?
Assuming your curves are ellipses, where are their foci?
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.
What happens when you drag point F outside of the circle?
It's time for a proof. How can you show that the shape outlined by the crease lines is an ellipse when point F is inside the circle? In the construction below, you'll see that radius AC intersects the crease line at point B. As you drag point A around the circle, notice that point B traces the curve in the picture. Point B is the tangent point on the crease.
Can you prove that point B traces an ellipse? Two hints: First, look for some congruent triangles. You'll need to add a segment to find them. Then, prove the curve is an ellipse by showing that FB + CB is constant.
In the second interactive model on this page, you may have noticed that when you dragged point F outside the circle, the crease pattern formed what looked to be a hyperbola.
Proving this conjecture isn't too different than the case of the ellipse. In the interactive model below, point F sits outside the circle. Point B is the intersection of the crease line with the line passing through A and C. Drag point A around the circle and watch point B trace what looks to be a hyperbola.
To construct a proof, you'll need the distance definition of a hyperbola: the set of points such that the difference of the distance from each point to two fixed points (the foci) is constant.
As before, look for a pair of congruent triangles.