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Triangle centres?

I was already aware that the "centre" of a triangle can be defined in a number of ways, e.g. the incentre (the centre of the inscribed circle), circumcentre (the centre of the circle that passes through the vertices) and the centroid (the "centre of mass" or balance point). But what about the point from which the sum of the distances to the vertices is minimised? I'm not even sure whether there is always a unique point which has this property.

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