A "sphere-some" foursome

Geometry Level pending

Suppose you have four non-coplanar points.

How many spheres can be found such that all four points lie on its surface?

6 1 3 2 0 It depends An infinite number 4

1 solution

Geoff Pilling
May 19, 2017

Call the non-coplanar points, $A, B, C$ , and $D$ .

Now let the locus of all the points that are the same distance between $A$ and $B$ be the plane $P_1$ .

Let the locus of all the points that are the same distance between $B$ and $C$ be the plane $P_2$ .

And, let the locus of all the points that are the same distance between $C$ and $D$ be the plane $P_3$ .

Now, the center of the sphere must lie on all three planes, and since the points are non-coplanar, these three planes intersect at exactly one point. Therefore, exactly $\boxed{1}$ such sphere can be found.

The question needs to be rephrased. As it stands, it calls for yes/no answer rather than the number of spheres that can satisfy the condition.

Marta Reece - 4 years ago

Thank you for your comment... Somehow I had updated it but forgotten to save it. :-P

Geoff Pilling - 4 years ago

A "sphere-some" foursome

1 solution

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