Pythagoras' Theorem without Matrices?

  Given the Pythagoras’ theorem  xx+yy= rr, one may derive this using 2x2 matrices, i.e. considering a 2x2 matrix which transforms (r,0) column (or (0,r)) into (x,y), One may then transpose the matrix with (r,0) row multiplied by the matrix transposed multiplied by the matrix multiplied by (r,0) column to find that xx+yy=rr. This is common knowledge based on a rotation matrix.

    In this note, we ask if one may obtain xx+yy=rr without using matrices. We first suggest that r maps into a set of x,y values if one considers moving a radius on a circle from (0,r) to (r,0). Given that r is a  single number, each corresponding pair (x,y) must map into this same single number. Thus, one seeks a mapping of two numbers, x,y into one number r. We then consider that one may x or -x and y or -y and suggest that one requires an even power of x and y with no x y mixing (i.e. no x multiplied by y). This leads to the sum of:   x(power 2) + x(power 4) +... + y(power 2) + y(power 4) + … as a possible solution. If one notes that (x,y) maps into { .5(x+y), .5(x-y)} with no information lost and argues for no mixing of x,y, then only xx+yy as a general formula delivers this property. We thus suggest that xx+yy=rr is the way to map (x,y) into r. 

   It is possible that this approach or one similar has already been presented in the literature due to the interest in Pythagoras’ theorem.