# Equidistant lines of two concentrical squares

Another math homework question:

There are two concentrical squares, ABCD and EFGH. AB = 5 and EF = 1 .

Draw the equidistant lines of those two squares. Clearly state how you do this.

My first approach was to simply draw a square inbetween the smaller and the bigger square with a side size of 3. But the corners make me uncertain that I am doing it right.

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Are the sides of squares parallel? Moreover, there might be no equidistant line (which is infinite), but there surely will be some equidistant points (in general, their set would be a union of segments and fragments of parabolas). – dtldarek Mar 28 '13 at 11:49

I guess we should define the distance between two sets, a possible way of doing this (getting ideas from the distance between a set and a point) is

Given a metric space $(X,d)$ we define the distance between two subsets $A,B$ as $d(A,B) := \inf \{d(a,b) : a \in A, b \in B\}$.

So let A be the bigger square, and B the smaller one. Of course we are in a metric space, indeed we know the length of the edges of the two squares (i.e. we can define $d(A,B)$).

Therefore I believe that drawing a square, say C, (exactly) between them is correct, indeed the distance, for example, between the lower left vertex in A and the lower left vertex in C is bigger (by the Pythagorean theorem) than $d(A,C)$ (the same argument applies with B and C).

I think this is a matter of definitions, I may have misunderstood the problem!

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