The total area can be calculated:

Either by adding the area of the inner square and the area of the triangles:

c²+4⋅÷{a⋅b}{2}=A

Or by the length of the outer square squared:

A=(a+b)²

Clean up a bit:

A=a²+2⋅a⋅b+b²
c²+2⋅a⋅b=A

And thus follows:

c²+2⋅a⋅b=a²+2⋅a⋅b+b²
c²=a²+b²