Pythagoras stated that the Hypotenuse (c) of a right triangle is:

c² = a² + b²

PROOF:

Given the Right (Rt) Triangle ABH with hypothenus, (c), and sides (a) and (b)
• From Rt. Triangle ABH, construct a large Rectangle ACEG, with each side equal to (a + b); and, a smaller Rectangle HBDF, with each side equal to (c). Thus, within Rectangle ACEG is the smaller Rectangle HBDF and four (4) Rt. Triangles: ABH, BCD, DEF, and FGH:
  The Area of ACEG is:
  1. Area = Length x Width; Area = (a + b)² = a² + 2ab + b²,
  2. the Area of ACEG is also equal to the Area of the smaller Rectangle HBDF plus the Areas of the four (4) Rt. Triangles:
     • Area of HBDF = Length x Width = (c) x (c) = c²,
     • the Area of each Rt. Triangle = 1/2((a)(b)): For the four = 4(1/2((a)(b)) = 2ab,
     • Thus, the Area of ACEG is also equal to: c² + 2ab.
  3. The Area of ACEG is:

     Area ACEG: a² + 2ab + b² = c² + 2ab

     a² + b² = c²