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?
In a right triangle with legs $a,b$ and hypothenuse $c$:
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 $a+b:$
$A= 4(1/2)ab + c^2$, where $(1/2)ab$ is the area of one triangle (why?).
$a^2+b^2 = c^2$.