The area of the small square = (a+b) 2 = a 2+2ab+b 2.

The area of the big square = c 2

The area of the four triangles = 1/2ab×4=2ab.

∫ large square area = large square area+four triangular areas

∴a^2+2ab+b^2=c^2+2ab

Simplified to a 2+b 2 = c 2

(2)

The area of the big square = c 2

Area of small square = (a-b) 2 = a 2-2ab+b 2

The area of the four triangles = 1/2ab×4=2ab.

∫ large square area = large square area+four triangular areas

∴c^2=a^2-2ab+b^2+2ab

Simplified to a 2+b 2 = c 2

(3)

Trapezoidal area =1/2 (a+b) 2 =1/2a2+ab+1/2b2.

The area of three triangles =1/2ab× 2+1/2c2 = ab+1/2c2.

Trapezoidal area = area of three triangles

∴ 1/2a^2+ab+ 1/2b^2=ab+ 1/2c^2

Simplified to a 2+b 2 = c 2