See the drawing at the attached file, the sides of the triangle are 1,1,√2

Ιwant a geometric relation for √2, i.e. something like that:

a/1=b/a=>b=a^2

and thus

√2=1+a+b+c+...=1+a+a^2+a^4+a^8+a^16+...

But a/1=b/ais probably wrong. Can you proove it if it’s correct?

Any other similar relation I could get?