The Pythagoras' theorem from a 1D Segment via Rotation

How a right triangle (and the Pythagorean identity) arises from an isometricrotation of a 1D segment. This short note does not claim a new theorem.It repackages three classical ingredients into a single, rotation-based construction:1) Rotation as an isometry (distance-preserving transformation),2) Thales' theorem (a right angle subtends a diameter),3) Pythagoras' theorem and, via an altitude decomposition, the Law of Cosines.With the notation used in this document, the segment AB has length s and plays therole of the hypotenuse. Rotating the endpoint B around the midpoint O moves B toB' on the circle with diameter AB, ensuring that triangle A B B' is right-angled at B'.The Pythagorean identity then follows directly, and the Law of Cosines arises bydecomposing a general triangle with an altitude into two right triangles.Intended use: a compact educational pathway from isometries to Pythagoras andthe Law of Cosines, with consistent notation and a transparent limiting case (h = 0).