Geometric Fallacy Proof:
Two circles with different radii have the same circumference length.
FIND THE ERROR IN THE FOLLOWING PROOF:
Construction: Draw two concentric circles (with the center O) of different radii. Mark the radius AO; it intersects the internal circle at the point
C. Roll the external circle (in the direction shown) along the tangent AB
until it makes exactly one rotation.
We have:
The center O moves to the point D.
The point A moves to the point B.
The point C moves to the point E.
Proof:
(1). Since the external circle makes exactly one rotation, the internal circle will make exactly one rotation as well. So, the length of the circumference
of the outer circle is equal to AB, and the length of the circumference of the
inner circle is equal to CE.
(2). OA = DB (radius of the external circle), and OA k DB (both OA
and DB are perpendicular to the tangent AB), so AODB is a parallelogram,
and OD = AB.
(3). OC = DE (radius of the internal circle), and OA k DB (by(2)), so
OCED is a parallelogram, and OD = CE.
(4). From (2) and (3), AB = OD = CE. Then, by (1), the lengths of the
circumferences of these two circles are equal.