maximum radius inside square of 2 identical circles.

Question

this is my first question on this site. I want to ask a question which is related to geometry. There is a square and 2 circles which have same radius inside in it. Now, question is what is the maximum radius of each circle? Can I get it if I put both centers on the diagonal of the square but I am not able to proof it? Please give the logic also proof for the same. Thanks

Answer 1

Let's say we have placed two circles of radius $r$ inside an unit square whose interior are disjoint from each other.

Since the circles lie inside the square, their centers have to be at least $r$ away from the sides. The centers are constrained to lie inside a smaller square of side $1-2r$.

Since the interior of the circles are disjoint, the distance between the two centers is at least $2r$.
Since the distance between any two points in the smaller circle is at most $\sqrt{2}(1-2r)$, we have:

$$2r \le \sqrt{2}(1-2r)\quad\implies\quad r \le \frac{1}{2+\sqrt{2}}$$ It is easy to verify this bound is achieved by placing the circles along the diagonal. This means the configuration you propose is indeed the optimal one.