Pythagoras' theorem

In impedance triangle ABC, in which quadrant is angle g, is:
            



c2 = a2 + b2

       2nd power of length of hypotenuse in impedance triangle is equal to sum of 2nd power of lengths of its rest sides.


Mathematical evidence by Law of cosine:

       As evidence of Pythagoras' theorem is enough to use equation: c2 = a2 + b2 - 2 a b cos g    from Law of cosine and substitute g = p / 2 .

Evidence of inverted Pythagoras' theorem:

       We'll make evidence by dissension: Let's predict that c2 = a2 + b2, while angle opossite to side c isn't quadrant. We can construct triangle ABC1 so that:
       |AC1| = |AC|,
       |angle AC1B| = 90o
      Triangle ABC1 is impedance with quadrant opossite to side c - for lengths of its sides b, c, a1 is true when wee say Pythagoras' theorem that c2 = a2 + b2. If left sides of equations are equal then also right sides must be equal so that means that:
       a12 + b2 = a2 + b2
We'll substract b2 : a12 = a2
       a1 and a are positive then we have: a1 = a - triangles ABC and ABC1 are identical by sss (side, side, side) but aren't identical in matching included angles ACB and AC1 - dissension. Assumption that angle ACB is quadrant is wrong. That means that if c2 = a2 + b2 is true then this triangle is impedance with quadrant opposite to side c - QED.

Evidence by similarity :

We'll use Euklides' theorem:
      

a2 + b2 = c.ca + c.cb = c ( ca + cb ) = c2 - > c2 = a2 + b2.