The hunting horn

Two semicircles and four tangents. What’s the area of the orange rectangle?

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The rectangle area is 8.

Calling the small radius r and the large one R, I have added a few dimensions to the diagram. Reflecting the semicircles in the diameter, we obtain circles and we can apply the Intersecting chords theorem. This gives 2r*2(R-r)=4*4. So 2r(R-r)=8. But this is exactly the area we need.

Visual solution

David Andriana kindly provided a visual solution to The hunting horn:

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