Note that, if we take the points A,B',C,A',B,C' (in that order) as the vertices of a hexagon, then the line pairs (AB', A'B, etc) are pairs of opposite sides of the hexagon.
Thus the theorem can be stated as
The points do not have to be distinct. For example, if we have B'=A, then the result holds with the "chord"
AB' being the tangent at A.
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