[gowld]

Geometry pictures are absolutely not "rigorous definitions". For 1 thing, pictures of lines aren't finite in extent, and pictures of points are finite in extent.

Some of Euclid's proofs are wrong, because they relied on those non-rigorous picture definitions.

* Book I, Proposition 1 (constructing an equilateral triangle)

* Book I, Proposition 4 (Side-Angle-Side triangle congruence).

* Multiple theorems throughout the Elements rely on the visual "betweenness" of points.