To construct an ellipse, we start with two points, A and B, to use as foci. We then create two line segments of equal length, AF and BH. The next step is to construct point E on segment AF and G on segment BH, such that the length of AE is always equal to the length of HG. These segments have been darkened in the applet.

Since AE = GH, we know that AE + BG = AF = BH.

The next step is to construct two cirles. The first has center A and passes through E, and the second has center B and passes through G. Notice that when E moves along the segment AF, the two circles chance radius, but their radii maintain a constant sum equal to the length of AF and BH.

The final step is to consider the intersections of these circles, which we will call P and Q. Since these points lie on both circles, PA = QA = AE and PB = QB = BG. Therefore, the sum of the distances from A to P and from B to P will equal the sum of AE and BG, which is the length of AF (and the length of BH).

This sum is a constant, and we will therefore expect the intersections P and Q to trace an ellipse as E and G move along AF and BH, respectively.

To use the applet, click and drag point E along the line segment AF. A and F can be repositioned by clicking and dragging, and the trace marks can be cleared by clicking the red X at the bottom of the window. Observe the changes when the distance between A and B increases and decreases, and when the length of AF increases and decreases. What appears to happen when A and B are equal?