Inscribed Squares Problem -- Theorem F

The Inscribed Squares Problem

THEOREM F: If T is any triangle and if J is a simple closed curve in 3-dimensional space R³ that is differentiable (has a tangent line) at at least one point, then J has an inscribed triangle similar to T.

Proof: Let the curve J in R³ and the triangle T be given. Let P and Q be points on J near the place where J is differentiable. Then since J is reasonably close to its tangent line at this point, the curve is relatively flat near P and Q.

Let s be a shortest side of triangle T and let C be the set of points X in R³ such that triangle PQX is similar to T with side PQ corresponding to side s. Then C is a circle surrounding the segment PQ, and since J is close to its tangent line, this cirlcle will also surround J -- in other words, C is linked with J if P and Q are close enough together.

Now move P and Q along J until they are at maximum distance apart:

The circle C is now unlinked from J! To see this, note that for any point X on C no point of J meets the ray PX beyond X (because the distance PX is at least as great as distance PQ which is the maximum distance possible between points of J). So, the union of these rays forms a "punctured shield" with C as its boundary that doesn't touch J. (See the figure below.)

Since C began as a circle linked with J and ended up unlinked from J, it must at some time have intersected J. And at the moment that intersection occured we had a triangle similar to T inscribed in J.

Note: This proof can be adapted for dimensions higher than 3 -- ie: it is true that any simple closed curve in Rⁿ (for any n greater than 1) must have an inscribed triangle similar to any given triangle. In the higher dimensional version, C is a shpere of dimension n-2 that goes from being "linked" with J to "unlinked" from J.