# Thread: Inscribing Largest Square in Circles' Intersection

1. ## Inscribing Largest Square in Circles' Intersection

The following is a problem from curiosity, since I was looking at other inscription cases.

If I have two equally sized circles with centers P and Q with P != Q with radius r, and their intersection is not empty nor a single point, what is the largest possible square that can be constructed within that intersection?

I started off by drawing the line segment PQ as well as the perpendicular bisector passing intersection points A,B. Intuitively, I can draw a square with one of the diagonals being min(PQ,AB) but this doesn't seem to be the largest. My intuition after that is that I can make a square bigger if I choose opposite points on the arc between PA QB and PB QA respectively, but I can't prove it.

Suggestions?

2. ## Re: Inscribing Largest Square in Circles' Intersection

Do you include varying the distance between their centers, or is that fixed?

If it's fixed, there is only one square that can be inscribed in the intersection.

r = 1;

Manipulate[
Show[{ParametricPlot[{{r Cos[q], r Sin[q]}, {r Cos[q] + d,
r Sin[q]}}, {q, 0, 2 \[Pi]}],
Graphics[
Line[{{r Cos[\[Pi] - q] + d, r Sin[\[Pi] - q]}, {r Cos[q],
r Sin[q]}, {r Cos[-q], r Sin[-q]}, {r Cos[\[Pi] + q] + d,
r Sin[\[Pi] + q]}, {r Cos[\[Pi] - q] + d,
r Sin[\[Pi] - q]}}]]}], {q, 0, \[Pi]/4}, {d, 0, 2}]

3. ## Re: Inscribing Largest Square in Circles' Intersection

I don't have Mathematica, unfortunately, but given your point that only one square can be inscribed in the intersection when the distance between their centers is fixed, this makes the question easier. Thank you.