I recently had a discussion about the Pythagorean theorem and how to understand it better. I realized that the role of the circle as a fundamental element is often underrated. You can hardly find any proof where the right-angled triangle is placed within a circle to support the reasoning. Instead, the triangles are usually presented in a plane without reference. However, when you position the triangle inside a circle, it becomes much easier to visualize how the relationship holds as the right-angle vertex moves along the circle.
The key idea to understand is that in the following image, for a unit square, the green and blue areas are always equal to cos²(2α), while the orange and red areas correspond to sin²(2α). It is evident that the square, with an area of 1, is the sum of its left and right sections.
By opening the GeoGebra resource [Pythagoras proof in a circle (opens a new tab)], you can slide the right-angle vertex (red point) anywhere along the upper half of the circle.
