A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.

To prove the Pythagorean Theorem using the described figure with four right-angled triangles and an inscribed square, we can follow these steps:

1. Understand the Components: In your figure, each right triangle has a height \( a \) and a base \( b \), with the hypotenuse labeled \( c \). The outer square is formed by placing these triangles at each corner, effectively using the right angles to construct the square's corners.

2. Identify the Dimensions of the Square: The sides of the outer square are composed of two segments \( a \) and \( b \). Since there are two parts to each side of the outer square, the total length of each side can be represented as \( a + b \).

3. Relate the Inner Square: The inner square, whose sides are labeled \( c \), is created by connecting the vertices where the legs of the triangles meet. The vertices of the inner square divide the sides of the outer square into segments of lengths \( a \) and \( b \).

4. Calculate the Area of the Outer Square: The area of the outer square, which has a side length of \( a + b \), can be computed using the formula: \[ \text{Area}_{\text{outer}} = (a + b)^2 \]

5. Calculate the Area of the Four Triangles: Each triangle has an area calculated as: \[ \text{Area}{\text{triangle}} = \frac{1}{2} \times a \times b \] Since there are four such triangles, their combined area is: \[ \text{Area}{\text{triangles}} = 4 \times \left( \frac{1}{2} \times a \times b \right) = 2ab \]

6. Calculate the Area of the Inner Square: The area of the inner square, built from the hypotenuses of the triangles, is: \[ \text{Area}_{\text{inner}} = c^2 \]

7. Relate the Areas of the Figures: The area of the outer square should equal the area of the inner square plus the area of the four triangles: \[ (a + b)^2 = c^2 + 2ab \]

8. Expand and Rearrange: Expanding the left side, we have: \[ (a + b)^2 = a^2 + 2ab + b^2 \] So, substituting this back into our equation gives us: \[ a^2 + 2ab + b^2 = c^2 + 2ab \]

9. Subtract \( 2ab \) from both sides: \[ a^2 + b^2 = c^2 \]

This final equation, \( a^2 + b^2 = c^2 \), is the Pythagorean Theorem. The proof effectively shows that by considering the areas of the geometric shapes defined by the right triangles and square, we can derive the fundamental relationship between the lengths of the sides of a right-angled triangle.