Pythagorean Theorem: MYTH or FACT ?

Let ABΓ be a right triangle. Let Kx be the perpendicular bisector of AΓ and By be the angle bisector of

B. Then we bring OM and ON perpendicular to AB and BC respectively.

So, AOM and AON are right triangles and:

i)OA=OΓ(from the perpendicular bisector property)

ii)OM=ON(from the angle bisector property)

So, AOM=AON =>AM=ΓN

Then equally we can proove that the reight triangles BMO and BNO are equal because

i)BO=BO

ii)OM=ON

So, BMO=BNO => BM=BN

Because of AM=ΓN and BM=BN if we substract these by parts..

AM - BM = ΓN - BN =>

So, AB=BΓ

Finally we have AB^2 +AΓ^2>AB^2  => AB^2 +AΓ^2>BΓ^2 .

This contradicts to the result of the pythagoerean theorem..What is your opinion about this proof?