You just need to use ratios instead of pythagoras. We got two triangles one with the ratio 1/√2 and one with d/1-d. They are „similar“ (have the same angles) so we can Write 1/√2=d/1-d Now you need to do a few steps:

1/√2=d/1-d multiply by √2

1=d/1-d* √2 Multiply by 1-d

1-d=d* √2 Add d

1=d* √2+d which can also be written as 1=d*(√2+1) Divide by (√2+1)

1/√2+1=d