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Nice fudge!
It is, of course, this statement that is incorrect ...
"If we take this process to its limit, we take our path to the path of the arc AC."
As the number of sub-triangles approaches infinity, it is the sum of the hypotenuses of each sub-triangle that approaches the length of the arc AC.
The hypotenuse of a right-triangle is always less than the sum of the other two sides. Therefore, taking the limit, the arc AC < AD + CD
or arc AC < r, the radius of the circle.
Regards, Rob.
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