Visual Proof of the Pythagorean Theorem

Overview

A concise visual demonstration that the areas of the squares constructed on the legs of a right‑angled triangle sum to the area of the square on the hypotenuse, i.e. $a^{2}+b^{2}=c^{2}$. The animation builds a right triangle, attaches squares on each side, then rearranges the two smaller squares to exactly fill the larger square.

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Phases

#	Phase Name	Duration	Description
1	Setup	~4s	Draw a right‑angled triangle with legs of lengths $a$ (horizontal) and $b$ (vertical). Construct outward squares on each side, labeling their side lengths $a$, $b$, and $c$.
2	Highlight Areas	~5s	Fade in a translucent overlay on each square to emphasize its area. Show the expressions $a^{2}$, $b^{2}$, and $c^{2}$ as simple area labels placed inside the respective squares (no extra text beyond the formulas).
3	Cut & Rearrange (Part 1)	~8s	Animate cutting the two smaller squares into a few geometric pieces (e.g., two right‑triangular halves and a central rectangle) and slide them toward the large square while keeping orientation consistent.
4	Cut & Rearrange (Part 2)	~8s	Complete the movement so that the pieces exactly fill the large square, demonstrating that the combined area equals $c^{2}$. A brief pause highlights the perfect fit.
5	Conclusion	~3s	Fade out the cut lines, leaving a solid square of side $c$ with a subtle glow, reinforcing the equality $a^{2}+b^{2}=c^{2}$.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (animation)             │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional, shows current phase)    │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Right triangle, three attached squares, cut‑and‑slide pieces, final assembled square	Central visual focus; occupies most of the frame
Caption	Short phase title (e.g., "Setup", "Rearrange")	Small persistent footer; helps viewers follow the sequence

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Notes

• Assumed right‑triangle legs $a$ and $b$ are of moderate length so that all squares fit comfortably within the frame.
• Used a minimal number of pieces (four) for the rearrangement to keep the animation clear and under 30 seconds total.
• No additional explanatory text is shown; the visual transformation itself conveys the proof.
• All transitions are smooth fades or linear slides to maintain pedagogical pacing.
• The specification assumes a single Manim Scene class.