In the interactive model below, AB is equal in length to DE, and AE is equal to BD. Drag point E and observe the curve traced by point C and its reflection, point F.
Now change the lengths of AB and AE by dragging the endpoints of the horizontal red and blue segments in the upper left corner. Then clear the trace by clicking on the red 'X' in the bottom right corner. Drag point E again and notice how point C's path changes.
Draw the above picture on a piece of paper and add segment AD to the illustration. Can you prove that triangle ADB is congruent to triangle DAE?
Now prove that triangle ACB is congruent to triangle DCE.
Use the above congruence to prove that point C traces an ellipse. Hint: Can you show that AC + CB is a constant?
Compare this linkage to the Congruent Triangles construction for a hyperbola.