= Solution
A <Hadamard space> is a complete <CAT(0) space>. A useful equivalent definition, which makes the remaining assertions consequences rather than assumptions, is a <complete metric space> in which every pair $x,y$ has a midpoint $m$, with $d(x,m)=d(y,m)=d(x,y)/2$, satisfying
$$
\boxed{d(z,m)^2\leq\frac12d(z,x)^2+\frac12d(z,y)^2-\frac14d(x,y)^2
\quad\text{for every }z.}
$$
This is the nonpositive-curvature midpoint inequality. Euclidean <Hilbert spaces> satisfy it with equality in the parallelogram formula. The comparison argument in (c) shows that this definition gives the usual CAT(0) triangle condition.
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