9873

Drawing an Oval with a String and Three Nails

It is commonly known that you can draw a circle using a pencil, a nail, and a loop of string: moving the pencil around the nail while holding it taut against the loop maintains a constant radius around the nail, and the resulting figure is a circle with the nail at its center.
A similar procedure using a pencil and string with two nails maintains a constant sum of distances from the two nails, and the resulting figure is an ellipse with each nail at one focus of the ellipse.
Carrying out the same process with three nails results in an oval or egg shape.

SNAPSHOTS

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DETAILS

Snapshot 1: the oval is made up of six elliptical arcs, representing the six regions formed by extending the line segments joining each nail (not counting the triangle formed by the three nails, in which none of the arcs fall)
Snapshot 2: if the three nails fall on the same line, or any pair of nails fall on the same point, the result is an ellipse; just as all nails falling on the same point results in a circle
Snapshot 3: if there is no slack in the loop of string, the figure is a triangle
M. Gardner, "Fun with Eggs, Part II," The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications, New York: Copernicus, 1997 pp. 59–66.

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