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There were a couple of awesome methods in Avenue for ArcView:

point=polyline.Along(percentage)
percentage=polyLine.pointPosition(point)

Don't laugh, but one of the reasons I updated to 10 from 9, was new method for Geometry

point=polyline.positionAlongLine (value,{use_percentage})

that is equivalent of 1st Avenue method (wrong name in my opinion anyway).

Does an equivalent of the 2nd Avenue method exist in ArcPy with ArcMap 10.2?

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Yes, it does exist. Use polyline.measureOnLine(point_geometry).

measureOnLine (in_point, {as_percentage})

Returns a measure from the start point of this line to the in_point.

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