Proof of Guogu or Pythagoras' Theorem


The most well-known theorem in Euclid Geometry is about the right-angle triangle, commonly attributed to Pythagoras.
Chinese mathematicians have known about this for a long time. A proof of the theorem is given in a mathematics book called "Zhoubi Suanjian" , as well as in a textbook called "Mathematics in Nine Chapters" , both of which have never been out of print in China.

Theorem states that:
a2 + b2 = c2

where  a  and  b  are the two sides and  c  is the hypotenus of a right-angle triangle.

Proof as given in Zhoubi Suanjian , Chap. 1.

Draw a rectangle with two sides of a and b(say the lower left).
Draw a diagonal of this rectangle (call it c).
Draw a square using the diagonal as its side (shown in red).
Replicate the original rectangle (marked by the red dot) three times around the red square.
Area of red square = Areas of 4 triangles + Area of small square.
the hypotenus; This is easily reduced to:
(Let  a  and  b  be the two sides and  c the diagonal)
4 (a b / 2) + ( b - a )2 = c 2
2 a b + b2 - 2 a b + a 2 = c 2
a2 + b2 = c2

Note: The diagram on the right is a reprint from an old edition with color added.
The diagram on the left is a copy of the original diagram with the grids removed.
Ref: Zhoubi Suanjian Complete text - In Chinese (BIG5 code)
  Proofs of Pythagoras' Theorem using Translations