Q:

Pythagorean Theorem - Incomplete proof

The image below is supposed to prove Pythagorean Theorem (without words).  However, I see that it is assuming that the square of side $c$ can be inscribed in the square of side $a+b$ and would result in the $4$ triangles of equal area.  This fact is only assumed, so would the proof be considered incomplete? 

A:

Theorem :  
In a right triangle with legs $a,b$ and hypothenuse $c$:
$a^2+b^2=c^2.$
Geometric proof:
Consider a square of side length $a+b$ (drawing). 
Partition the sides into $a$ and $b$ as shown. 
Each of the $4$ corner triangles formed are congruent by $SAS$, i.e. $a,b$, and right angle between. 
Hence the quadrilateral formed by connecting the partition  points has $4$ equal sides $c$, a rhombus. 
The adjacent sides of the inner quadrilateral are perpendicular (why?). 
Hence the inner quadrilateral is a square. 
Area of outer square of side length $a+b:$
$A:= (a+b)^2.$
Also: 
$A= 4(1/2)ab  + c^2$, where $(1/2)ab$  is the area of one triangle (why?). 
Finally:
$a^2+b^2 = c^2$.

