Euler’s Quadrilateral Theorem. I. A brief new Proof that is Physically Guaranteed to Generalize. Alias Eigenclustering Length-Exchange Theorems. I. P_2. and bent-P_3

This is v5, clearing out a 2018-era running write-up error. For now the error continues to run in Articles II to IV; this sentence shall be removed once we have got round to fixing that. It affects none of those Articles’ actual results!

This is v4, including a first comparative comment with previous proofs of this Theorem by others.

This is v3, with a reference added and figures finalized. Including removing an obvious typo from Fig 1.

This is the v2, correcting 2 typos in the v1 as pointed out by Everard. The vector in equation 7 had its components the wrong way up as regards being in the right basis to give the fundamental triangle matrix F rather than some column permutation thereof. Also the RoG Lemma (B) in Fig 2 was missing its square, without which it would be neglecting one factor of the system’s total mass.