This website contains the instructions for creating and proving both an ellipse and a hyperbola using Geometer's Sketchpad.

Ellipse

Under display, choose autoshow labels for points only. Be sure that precision for distances is hundredths.

1. Construct a circle in the center of the screen (directrix).
2. Use the point tool to create a point inside of the circle.
3. Select the circle and construct a point on object.
4. Construct a segment from the point inside the circle to the point on the circle.
5. Select the segment, and construct a point at midpoint.
6. Select the segment and the midpoint, and construct a perpendicular line.
7. Select the line through the midpoint, and choose trace line.
8. Select the circle and the point on the circle. Under edit, select action button and animate.
9. Double click on the animate button. What do you see?
10. How is an ellipse defined as a locus of points? 
11. Construct a line through the center of the circle and the point on the circle.
12. Construct a point at the intersection of the new line with the perpendicular bisector created above.
13. Select the perpendicular bisector, and unselect trace line.  Select the intersection point, and select trace point. (It may also be helpful to change the color of this point.)
14. Double click the animate button. What do you see now?
15. Construct a segment from the point of intersection to the point inside the circle. 
16. What do these triangles tell you? How do they prove you have drawn an ellipse?

Hyperbola

Under display, choose autoshow labels for points only. Be sure that precision for distances is hundredths.

1. Construct a circle in the center of the screen (directrix).
2. Use the point tool to create a point outside of the circle.
3. Select the circle and construct a point on object.
4. Construct a segment from the point outside the circle to the point on the circle.
5. Select the segment, and construct a point at midpoint.
6. Select the segment and the midpoint, and construct a perpendicular line.
7. Select the line through the midpoint, and choose trace line.
8. Select the circle and the point on the circle. Under edit, select action button and animate.
9. Double click on the animate button. What do you see?
10. How is a hyperbola defined as a locus of points?
11. Construct a line through the center of the circle and the point on the circle.
12. Construct a point at the intersection of the new line with the perpendicular bisector created above.
13. Select the perpendicular bisector, and unselect trace line.  Select the intersection point, and select trace point. (It may also be helpful to change the color of this point.) 
14. Double click the animate button. What do you see now? 
15. Construct a segment from the point of intersection to the point outside the circle. 
16. What do these triangles tell you? How do they prove you have drawn a hyperbola?