Magic Squaresproof atlas
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General proof chapter

Red lemma: three squares in arithmetic progression

A red triple means the linear condition U + W = 2V on three cells whose values are all perfect squares.

01r=−a2+2ab+b2,s=a2+b2,t=a2+2ab−b2r=-a^2+2ab+b^2,\qquad s=a^2+b^2,\qquad t=a^2+2ab-b^2
02r2+t2=2s2r^2+t^2=2s^2
03(U,V,W)=(k2r2, k2s2, k2t2)⟹U+W=2V(U,V,W)=(k^2r^2,\,k^2s^2,\,k^2t^2)\Longrightarrow U+W=2V
Substituting the triple into the corresponding Magic3 positions turns this linear equality into the required common coordinates E, x, and y.