Two flats P and Q in $R^n$ are parallel if and only if the (shortest)distance from any point in P to Q is the same.
Intuitively, a translation $T_v:T_v(x)=x+v$ should perserve paralle relation. That is, if P is parallel to Q, then $T_v(P)$ is also parallel to Q.
I tried using standard calculation using orthogonal projection but was unable to prove it.
specifically, I want to show that if $||x_1-P_Q(x_1)||=||x_2-P_Q(x_2)||$ then $||x_1+v-P_Q(x_1+v)||=||x_2+v-P_Q(x_2+v)||$ ,where $P_Q$ is the orthogonal projection to Q. I'm not sure how to do this.