Solution to Problem 177

A correct solution was received from Jeremy Light.  Partial solutions came from Adam Brown and Scott Hoffmeyer.  Further correct solutions were received from Ahmed AboAdham, Bill Webb, Alan O'Donnell, Nancy Schwarzkopf, Ron Welch, Juan Carlos Marivela, Al Zimmermann, Monty Gray.

The radius of the sphere is  (Ö5 - 1)/Ö3.

First, view the cones from above, as in Figure 1. Notice that the tips of the cones form an equilateral triangle with sides of length 2. Therefore, we can use trigonometry to find the distance from the point of one of the cones to the point where the sphere is tangent to the plane, shown as the length of one of the blue lines. This value is 2/Ö3.

Now view the picture from the side as in Figure 2. Note that the angle OAR bisects angle BAR, and that the tangent of angle BAR equals 2. By identity, the tangent of angle OAR equals (Ö5 - 1)/2. Thus, the radius, r, equals 2/Ö3 times (Ö5 - 1)/2, or (Ö5 - 1)/Ö3 as claimed.

Figure 1

Figure 2
Figure 1Figure 2

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ã2003 Alberto L. Delgado