Construction of circular arcs through vertices of a rectangle, tangent to the diagonals?

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I know that you can construct the tangent line to a circle, for example, but my question is in the reverse order:

If one draws a rectangle, can you use a compass and a straightedge to construct the circular arcs that pass through pairs of adjacent corners while being tangent to the rectangle's diagonals? Basically, can you make this shape?

(note: this image is an approximation, and is not the true construction I am looking for)

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