Visual Proof of the Pythagorean Theorem

Overview

A concise visual demonstration that the areas of the squares constructed on the legs of a right triangle combine to exactly fill the square constructed on its hypotenuse, illustrating the relationship $a^{2}+b^{2}=c^{2}$.

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Phases

#	Phase Name	Duration	Description
1	Intro	~2s	The right triangle appears centered, legs aligned horizontally (length $a$) and vertically (length $b$). The hypotenuse $c$ is highlighted.
2	Construct Squares	~4s	Squares are drawn outward on each side: a blue square of side $a$, a green square of side $b$, and a red square of side $c$. Side‑length labels $a$, $b$, $c$ appear on the corresponding edges.
3	Dissection & Rearrange	~5s	The two smaller squares are partitioned into a few geometric pieces (triangles and rectangles). Those pieces animate smoothly to fill the large $c$-square without overlap, demonstrating that the total area matches.
4	Reveal Equality	~2s	The equation $a^{2}+b^{2}=c^{2}$ fades in at the bottom of the screen, synchronised with the completed rearrangement.
5	Outro	~1s	All objects fade out, leaving a brief blank frame.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│              MAIN (primary visual)          │
│                                             │
├─────────────────────────────────────────────┤
│  Caption / label (step title or equation)   │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Right triangle, three squares, dissected pieces, animated transforms	Uses distinct colors for each square; minimal text (only side labels)
Caption	Current phase title (e.g., "Construct Squares") or the final equation $a^{2}+b^{2}=c^{2}$	Appears only when needed; fades in/out with its phase

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Notes

• Assumptions: The animation runs about 14 seconds total, uses smooth linear or ease‑in‑out transitions, and employs a single Manim Scene class.
• Side‑length labels are placed mid‑edge of each square; the equation appears in the caption area after the rearrangement.
• No additional explanatory text is shown; the visual steps themselves convey the proof.
• Colors: blue for the $a$-square, green for the $b$-square, red for the $c$-square; pieces inherit the color of their source square.
• The dissection follows the classic “four‑triangle” proof: each leg‑square is split into two right‑triangles and a rectangle, which together re‑form the hypotenuse‑square.
• All objects remain within the Main area; the Caption area is reserved for temporary labels and the final equation.
• The specification is designed for a single Scene; no separate sub‑scenes are required.