Question: how to proof Pythagoras theorem by using area theorem.
(Try to draw the diagram as described below)
Draw a square.
Inside this square draw another (smaller) square. It will be rotated a bit, such that each of the 4 corners of the smaller square touches a side of the larger square.
Label the sides of this (smaller) square as c.
The internal square will have divided each side of the external square into (probably) unequal sections, of lengths a and b, say.
The other three sides of the external square will also be divided into the lengths a and b.
Compare the areas in differrent ways.
The area of the external square is A1 = (a + b)^2
The area of the external square is also equal to the area of the internal square plus 4 times the area of the corner triangles.
Area of internal square is A2 = c^2
Area of 4 corner triangles is A3 = 4*(1/2)*ab
So, A1 = A2 + A3
i.e. (a + b)^2 = c^2 + 4*(1/2)*ab
a^2 + 2ab + b^2 = c^2 + 2ab
a^2 + b^2 = c^2