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Source. Published p. 361, Lemma 26 (published scan).

Statement. Given finitely many pairs $x_i,y_i\in\mathbb R^d$, with repetitions permitted, there is an $a\in\mathbb R^d$ satisfying $\|x_i-a\|=\|y_i-a\|$ for all $i$ if and only if every real relation $\sum_i c_i(x_i-y_i)=0$ satisfies $\sum_i c_i(\|x_i\|^2-\|y_i\|^2)=0$.

Complete proof. The distance equalities are equivalent to the linear system

$$2\langle x_i-y_i,a\rangle=\|x_i\|^2-\|y_i\|^2.$$

A solution makes the stated compatibility necessary by taking linear combinations. Conversely, under that compatibility the assignment $x_i-y_i\mapsto(\|x_i\|^2-\|y_i\|^2)/2$ extends linearly to a well-defined functional on the span of the differences: every relation is sent to zero. Represent this functional by inner product with a vector in that span, using an orthonormal basis. That vector solves all equations. $\square$

Bears on. #174.