Visual proof of the Pythagorean theorem using geometric decomposition.

You’ve uploaded an image that illustrates a geometric proof of the Pythagorean Theorem using area decomposition. Let’s break it down step by step.

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## 🎯 Goal:
Prove the Pythagorean Theorem:
> In a right triangle with legs `a` and `b`, and hypotenuse `c`,
> a² + b² = c²

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## 🔍 Understanding the Diagrams

There are two large squares, both with side length `(a + b)`. So each has area:

> (a + b)² = a² + 2ab + b²

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✔ Left Square (First Diagram)

This square is divided into:

- One yellow square of side `b` → area = b²
- One green square of side `a` → area = a²
- Two red right triangles, each with legs `a` and `b` → area of one triangle = `(1/2)ab`, so two = ab

So total area = a² + b² + ab

Wait — but we said the whole square is `(a + b)² = a² + 2ab + b²`.
That means we’re missing one more triangle? Actually, looking closely, there are four triangles!

Actually, let’s recount carefully:

In the left diagram, you have:

- 1 square of side `a` → area `a²`
- 1 square of side `b` → area `b²`
- 4 right triangles, each with legs `a` and `b` → total area = 4 × (½ab) = 2ab

✔ So total area = a² + b² + 2ab = (a + b)² — perfect match.

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✔ Right Square (Second Diagram)

This square is also side `(a + b)`, so area = (a + b)²

It’s divided into:

- One tilted blue square in the center with side `c` → area = c²
- Four right triangles around it, each with legs `a` and `b` → total area = 4 × (½ab) = 2ab

So total area = c² + 2ab

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## 🧠 Equating Both Areas

Since both large squares have the same total area `(a + b)²`, we can set their internal areas equal:

> Left square area = Right square area
> ⇒ `a² + b² + 2ab = c² + 2ab`

Now subtract `2ab` from both sides:

> a² + b² = c²

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## ✔ Conclusion

This visual proof demonstrates the Pythagorean Theorem by showing that the area of the large square can be decomposed in two different ways — one emphasizing the squares on the legs (`a² + b²`) and the other emphasizing the square on the hypotenuse (`c²`) — and since both total areas are equal, we conclude:

> a² + b² = c²

This is a classic and elegant geometric proof often attributed to ancient mathematicians like Bhaskara or even Euclid.

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## 📌 Final Answer:

> The Pythagorean Theorem is proven: a² + b² = c²

This follows from equating the total area of the large square computed in two different ways — once as the sum of the squares on the legs plus four triangles, and once as the square on the hypotenuse plus the same four triangles. Subtracting the common term (2ab) gives the desired result.

Let me know if you’d like to see an algebraic version or animation!