How to Prove Triangle Centers in Tetrahedra

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How would you prove the existence of triangle centers in tetrahedra, for example, the incenter, circumcenter, or centroid?

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By construction. For example the incenter, if it exists, is the point equidistant from all 4 faces. Construct it as the intersection of 3 of the 6 bisector planes, to be chosen so that the 3 corresponding edges do not meet in one vertex, and conclude that this intersection must lay at the same distance to all 6 faces (so that it automatically lies on the other 3 bisector planes as well).