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A point can be intersected by infinitely many lines. Two points intersect in only one line. Three points either intersect in a line or not at all. This is only considering two dimensions.
A diagonal in 2-dimensions, an edge or diagonal in 3-dimensions
Point : 0 dimensions (position only) Line: 1 dimension (length) Plane: 2 dimensions (length and width)
the middle point * * * * * In 2 dimensions: also any point on line forming the perpendicular bisector of the line segment. In 3 dimensions: the plane formed by the perpendicular bisector being rotated along the axis of the line segment. In higher dimensions: Hyperplanes being rotated along the same axis.