Benedetto Scimemi,  Paper-folding and Euler's Theorem Revisited,

Forum Geometricorum, 2 (2002) 93--104.

Abstract:  Given three points O, G, I, we give a  simple construction by paper-folding for a triangle having these points as circumcenter, centroid, and incenter. If two further points H and N are defined by OH = 3OG = 2ON, we prove that this procedure is successful if and only if I lies inside the circle on GH as diameter and differs from N. This locus for I is also independently derived from a famous paper of Euler, by complementing his calculations and properly discussing the reality of the roots of an algebraic equation of degree 3.

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