The Pythagoras' theorem from a 1D Segment via Rotation
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2026
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| author | Kaczorowski, Jarosław |
| author_facet | Kaczorowski, Jarosław |
| contents | <p>How a right triangle (and the Pythagorean identity) arises from an isometric<br>rotation of a 1D segment. </p> <p>This short note does not claim a new theorem.<br>It repackages three classical ingredients into a single, rotation-based construction:<br>1) Rotation as an isometry (distance-preserving transformation),<br>2) Thales' theorem (a right angle subtends a diameter),<br>3) Pythagoras' theorem and, via an altitude decomposition, the Law of Cosines.<br>With the notation used in this document, the segment AB has length s and plays the<br>role of the hypotenuse. Rotating the endpoint B around the midpoint O moves B to<br>B' on the circle with diameter AB, ensuring that triangle A B B' is right-angled at B'.<br>The Pythagorean identity then follows directly, and the Law of Cosines arises by<br>decomposing a general triangle with an altitude into two right triangles.<br>Intended use: a compact educational pathway from isometries to Pythagoras and<br>the Law of Cosines, with consistent notation and a transparent limiting case (h = 0).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18213256 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Pythagoras' theorem from a 1D Segment via Rotation Kaczorowski, Jarosław <p>How a right triangle (and the Pythagorean identity) arises from an isometric<br>rotation of a 1D segment. </p> <p>This short note does not claim a new theorem.<br>It repackages three classical ingredients into a single, rotation-based construction:<br>1) Rotation as an isometry (distance-preserving transformation),<br>2) Thales' theorem (a right angle subtends a diameter),<br>3) Pythagoras' theorem and, via an altitude decomposition, the Law of Cosines.<br>With the notation used in this document, the segment AB has length s and plays the<br>role of the hypotenuse. Rotating the endpoint B around the midpoint O moves B to<br>B' on the circle with diameter AB, ensuring that triangle A B B' is right-angled at B'.<br>The Pythagorean identity then follows directly, and the Law of Cosines arises by<br>decomposing a general triangle with an altitude into two right triangles.<br>Intended use: a compact educational pathway from isometries to Pythagoras and<br>the Law of Cosines, with consistent notation and a transparent limiting case (h = 0).</p> |
| title | The Pythagoras' theorem from a 1D Segment via Rotation |
| url | https://doi.org/10.5281/zenodo.18213256 |