CONO-SUR 1995 Problem 3

Let $ABCD$ be a rectangle with: $AB=a$ , $BC=b$ . Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$ ,the other is tangent to the sides $CB$ and $CD$ .

1. Find the distance between the centers of the circles(using $a$ and $b$ ).
2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

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