Danny’s Ellipse

In the early 1990s, Danny Vizcaino, a high school student at Monte Vista High School in California, wrote to Key Curriculum Press noting that Sketchpad did not come with a tool to draw an oval. Undaunted by this omission, Danny had built his own oval with the software and shared it with Key’s editors.

As shown in the interactive Web Sketchpad model below, Danny started by constructing two concentric circles with center at point A. He continued by adding a radius AD that intersected the smaller circle at point E. Danny then built a right triangle with hypotenuse DE whose base and height are parallel to the horizontal and vertical lines that pass through point A.

Drag point D and observe the trace of point F. The oval you form certainly looks like an ellipse, but is it? After receiving Danny’s construction, Key Curriculum issued a challenge to teachers and students in its newsletter: Could they prove that Danny’s oval was an ellipse?

The editors at Key received hundreds of letters and a potpourri of proofs demonstrating that Danny had indeed built himself an ellipse. Can you and your students devise one or more proofs of your own?

Danny’s technique for constructing ellipses turns out to be exceptionally handy because it does not require us to know the location of the ellipse’s two foci. We only need indicate the center of the ellipse and the lengths of its major and minor axes. If you click the arrow in the bottom-right corner of the web sketch above, you’ll see a neat animation of the words “Dynamic Geometry” that was built using Danny’s method.

If you’d like to explore other methods of constructing ellipses, check out my prior blog posts, The Congruent Triangle Construction and The Tangent Circles Construction. You’ll find these and many more conic section constructions in my book, Exploring Conic Sections with The Geometer’s Sketchpad.

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