Why the isometry $h(v)=Av+w$ is a rotation around $P$ when $\det A=1$

geometry linear-algebra proof-explanation isometry

If $v_0$ is a fixed point of $h(v) = Av+w$ then $h(v)$ can be represented as $h(v) = A(v-v_0) + Av_0 + w = A(v-v_0) + v_0$ where $A$ is a rotation matrix. But expression $A(v-v_0)+v_0$ is a rotation by matrix $A$ around $v_0$ because it represents translation of $v_0$ to the origin, rotation of the translated $v$ (which is $v-v_0$) by $A$, then translating it back to $v_0$. enter image description here

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