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Statement. If $z_1,z_2,z_3,z_4\in\mathbb R^2$ and $|z_1-z_2|=|z_3-z_4|>0$, there is exactly one orientation-preserving Euclidean isometry $g$ with $g(z_1)=z_3$ and $g(z_2)=z_4$.

Source. Mathialagan, published 2021 PDF, p. 10, Proposition 20. The printed statement omits $>0$, although its proof uses it. The nonzero condition is part of the application to the paper's energy.

Proof. A proper Euclidean isometry has form $g(x)=Rx+t$, where $R$ is a planar rotation matrix. Indeed, after subtracting $g(0)$, preservation of squared distances and the polarization identity preserve inner products; the images of the standard orthonormal basis then determine an orthogonal linear map. Preservation of orientation selects determinant $1$.

The two endpoint requirements force $R(z_2-z_1)=z_4-z_3$. Two equal-length nonzero vectors determine a unique rotation: their normalized directions specify its sine and cosine. This fixes $R$, and then $t=z_3-Rz_1$ is forced. Conversely these choices meet both requirements. If $R=I$, the motion is a translation, including the identity. Otherwise $I-R$ is invertible, since $\det(I-R)=2-2\cos\alpha>0$ for its angle $0<\alpha<2\pi$. Its unique fixed point is $(I-R)^{-1}t$, so it is a nonidentity rotation about that point.

If the common length were zero, arbitrary rotations followed by the forced translation would satisfy the requirements, explaining the qualification.

Application. Each positive-energy quadruple has one proper motion with $g(p_1)=q_2$ and $g(q_1)=p_2$, since the source segment $(p_1,q_1)$ and target segment $(q_2,p_2)$ have the same positive length. Partition these motions into translations and nonidentity rotations.

Verification scope. Verified within the independently reviewed Theorem 3 chain, retained in the final review; the corrected hypothesis and the motion classification are components of the living Theorem 3 record.

Bears on. Problem 661.