Pythagorean Theorem “Wordless Proof” Special Case: 1:3 Ratio

This example presents a wordless-style construction that visually demonstrates the Pythagorean Theorem in a particular special case: when a leg-to-hypotenuse ratio is fixed (e.g., 1:3), the relationship between the squares on the legs and the square on the hypotenuse becomes geometrically apparent. The key mathematics includes the Pythagorean Theorem, right-triangle side-length relationships via squared lengths, and how a constrained ratio guides a deterministic layout for area/shape equivalence. Construction highlights: it uses GeoGebra commands such as If (conditional control), Intersect (intersection points), Point, Line, Circle, CircleArc, Polygon, Midpoint, Translate, Rotate, and Angle. These are combined with geometric objects to build and reposition congruent or corresponding regions (via Midpoint/Translate/Rotate), yielding a clear visual correspondence consistent with the theorem. Suitable for high school geometry, classroom demonstrations, and exploratory learning of geometric proofs.