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Eulerizing a graph means to change the graph so that it contains an Euler circuit. To do this, we make use of Fleury's algorithm, which tells us that a graph with an Euler circuit in it has zero.
Take any triangle ABC. Construct the circumcenter, centroid, and orthocenter. These three points lie on the Euler line. Proof: We know that PY is perpendicular to AC, since it is the perpendicular bisector.