shows how the Pythagorean Theorem is equivalent to the Parallel Postulate. In my mind the parallel postulate was always linked to ideas of similarity, scaling and multiplication ( constructing similar triangles is effectively multiplying one line segment by another ). Does the equivalence to the Pythagorean theorem mean that the fifth dictates how area is to be defined in an Euclidean Geometry? Or that how area is defined determines the geometry?
The link
http://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml
shows how the Pythagorean Theorem is equivalent to the Parallel Postulate. In my mind the parallel postulate was always linked to ideas of similarity, scaling and multiplication ( constructing similar triangles is effectively multiplying one line segment by another ). Does the equivalence to the Pythagorean theorem mean that the fifth dictates how area is to be defined in an Euclidean Geometry? Or that how area is defined determines the geometry?