Question

show that there is one point whose coordinates don't alter due to rigid motion​

Answer 1

We need to show that exist x fixed point such that

x=Rx+x0⟹(I−R)x=x0

and thus that

for x0=0⟹x=0 (no translation)

or

for x0≠0(I−R) is invertible and x=(I−R)^−1 x0