Geometric Proof of the Pythagorean Theorem

Overview

A concise visual demonstration that the area of the square on the hypotenuse of a right‑angled triangle equals the sum of the areas of the squares on the other two sides. Viewers see a right triangle, the construction of the three squares, and a rearrangement that makes the equality evident.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Fade‑in a right‑angled triangle $\triangle ABC$ with right angle at $C$. Axes are hidden; a subtle highlight marks the right angle.
2	Construct Squares	~7s	Squares are drawn outward on each side: $\square$ on $AB$ (hypotenuse), $\square$ on $AC$, and $\square$ on $BC$. Each square is colored distinctly and labeled with its side length (e.g., $c$, $a$, $b$).
3	Area Decomposition	~5s	The large square on $AB$ is subdivided into four congruent right triangles (copies of $\triangle ABC$) and a central empty region. The two smaller squares are shown to fill exactly this central region when the triangles are rearranged.
4	Rearrangement Animation	~4s	The four triangles rotate and slide to the corners of the large square, leaving the central region. The two smaller squares slide into that region, perfectly covering it.
5	Conclusion	~3s	A final overlay displays the equation $c^{2}=a^{2}+b^{2}$ with a brief highlight, then all elements fade out.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (visual proof)         │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional) – step title or formula │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Right triangle, three squares, moving pieces during rearrangement	Central focus; occupies most of the frame
Caption	Short step label (e.g., "Construct Squares", "Rearrange") and final equation $c^{2}=a^{2}+b^{2}$	Appears only in phases 2‑5; omitted during Intro

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Notes

• Assumptions: The triangle is drawn with legs $a$ and $b$ of moderate length to keep the scene balanced; colors are chosen for clear contrast. No textual explanation beyond the final equation is used, adhering to the “avoid text unless necessary” rule.
• All animations are kept short to stay well under the 30‑second limit (total ≈22 s).
• The scene uses a single Manim Scene class; all phases are sequenced within that class.
• No additional sound, resolution, or technical rendering details are specified.