Please help me understand the definition of straight line given by Euclid. "A straight-line is (any) one which lies evenly with
points on itself."
This is how Euclid defines a straight line but I don't know what it really means.
Is this saying that any point picked on the straight line will have equal distance from each other?
What does he mean exactly by lies evenly?

Within the Euclidean system of geometry that is articulated in Book 1 of the Elements, the three fundamental symmetries for objects in two dimensions are reflection, translation and rotation. The idea that a straight line lies evenly between two points can be understood in terms of reflection. If a line A were to be transformed by reflection along an axis running along the two points to yield line B, then the line A would be straight if the reflection to B yielded a line that, when placed on top of A, every point on B lies evenly on A. If a line C had any curvature or deviation from what is straight, then the reflection would yield a line D that does not lie evenly on line C.
Note that this explanation of what it is for a line to be straight rests on a fundamental kind of symmetry--and so it conforms to modern notions of the kinds of group relations that are taken to be fundamental across different kinds of geometries (see Cayley and Klein). As such, Euclid's attempt to dig down to fundamental assumptions does seem to yield remarkable insights.
