How to Prove Pythagoras' Theorem - visual geometric proof using squares and triangles.

The image you've uploaded is a visual representation of one of the most elegant and classic geometric proofs of Pythagoras' Theorem. Let's walk through how this diagram helps prove the theorem.

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🔷 Pythagoras' Theorem Statement:

In a right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$):

$$
a^2 + b^2 = c^2
$$

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🔷 Understanding the Diagram

We see:

1. A right triangle with legs $a$ and $b$, and hypotenuse $c$.
2. Below it, a large square constructed using four copies of this triangle arranged around a smaller square in the center.

Let’s analyze this step by step.

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🔷 Step 1: Construct a Large Square

We form a large square with side length $a + b$. This means:

$$
\text{Area of the large square} = (a + b)^2
$$

This large square contains:
- Four identical right triangles (each with legs $a$ and $b$, hypotenuse $c$)
- One smaller square in the center, whose sides are equal to $c$

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🔷 Step 2: Calculate Area in Two Ways

#### ✔ Method 1: Directly from the large square
$$
\text{Total area} = (a + b)^2 = a^2 + 2ab + b^2
$$

#### ✔ Method 2: Sum of areas of parts inside

- Each triangle has area: $\frac{1}{2}ab$
- Four triangles: $4 \times \frac{1}{2}ab = 2ab$
- The inner square has side $c$, so its area is $c^2$

So total area:
$$
\text{Total area} = 2ab + c^2
$$

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🔷 Step 3: Equate the Two Expressions

Since both expressions represent the same total area:

$$
(a + b)^2 = 2ab + c^2
$$
$$
a^2 + 2ab + b^2 = 2ab + c^2
$$

Now subtract $2ab$ from both sides:

$$
a^2 + b^2 = c^2
$$

✔ This proves Pythagoras' Theorem!

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🔷 Visual Insight

The key idea is that:
- The large square can be split into four triangles and a central square.
- By calculating the area in two ways — once as $(a+b)^2$, and again as the sum of triangle and central square areas — we derive the identity $a^2 + b^2 = c^2$.

This is a proof by dissection, where the rearrangement of shapes shows equality of areas.

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✔ Summary

> Pythagoras' Theorem: In a right triangle,
>
> $$
> a^2 + b^2 = c^2
> $$

This proof uses geometry and area comparison to show that the sum of the squares on the legs equals the square on the hypotenuse.

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Let me know if you'd like an animation explanation or a different proof (like algebraic or similar triangles)!