Mean distance vs. distance to mean (L2 norm)

Question

Intuitively, I feel like the following should hold, but I fail to prove it:

$\lVert x^*- \frac 1k \Sigma_{i=1}^k w_ix_i\rVert_2 \overset ?= \frac 1k \Sigma_{i=1}^k w_i \lVert x^*- x_i\rVert_2$

Where $x_1,...,x_k, x^*\in\mathbb R^d$

I.e. is the distance to a (weighted) average point equal to the average of (weighted) distances to original points?

Answer 1

A simple counterexample: $d=1,k=2,x_1=0,x_2=2,x^{*}=1,w_1=w_2=1$.