The theorem of Pythagorus - 3 Proofs

The theorem says that in a triangle with one angle 90 degrees that the sum of the squares of the length of that two sides that form that 90 degrees will be equal to the square of the length of the side opposite of that 90 degrees angle.

Proof

See figure below and read the data on it

We started with the black triangle CGD. All other lines are constructed.

The red lines are all constructed so that they form squares.

In the triangles DEC and ADG the following can be observed.

  1. DE = DG
  2. AD = DC
  3. Angle ADG = angle EDC

So triangle DEC is congruent to triangle EDC

Therefore the area of triangle DEC = the area of triangle ADG

Area triangle DEC = 0.5 times DE times EF = 0.5 b2

Area triangle ADG = 0.5 times a2 times a

So b2 = a times a2

In the triangles DCI and BGC the following can be observed.

  1. BC = DC
  2. CI = GC
  3. Angle BCG = angle DCI

So triangle DCI is congruent to triangle BGC

Therefore their areas must be the same.

Area triangle BCG = 0.5 times BC times a1 = 0.5 times a times a1

Area triangle DCI = 0.5 times CI times HI = 0.5 c2

So c2 = a times a1

Therefore a times a1 + a times a2 = b2 + c2

So a(a1 + a2) = b2 + c2 but a1 + a2 = a

So a times a = a2 = b2 + c2

Case proven

>

See Proof 2 below

>

See Proof 3 below