In an arbelos, the distance between two points lying in a tangent line to the two smaller semicircles is 4.
What is it exact area?
Solution
The distance between the two tangent points is 4.
Then the vertical line segment with endpoints the point of intersection between the two smaller semicircles and the point of intersection between the line segment and the big semicircle has also length 4.
The four endpoints of the two line segments are concyclic, i.e., lie in the circle of diameter 4.
The area of the arbelos is equal to the area of the circle with radius 2.
So, the area of the arbelos is
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