



 
Hi Alexander, The answer depends on the dimensions of your rectangle. Start with a long, thin rectangle (whose length is in the horizontal direction and is at least twice its width); the two LARGEST congruent circles are those tangent to both long sides; if you want, you can take them tangent to one another at the center of the rectangle. Their common tangent is the vertical line through the center of the rectangle. They will be tangent also to the short sides, one on the right, the other on the left, when the rectangle's length is exactly twice the width. Now keep the width fixed but shrink the length. One circle will stay tangent at the top and left, and the congruent mate will be tangent on the bottom and the right; they will be tangent to one another at the center of the rectangle, and the common tangent will be a slanted line. Eventually the rectangle becomes a square (with length equal to width), and the common tangent will be the diagonal that runs from the bottom left corner to the top right corner. Chris  


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