The point inside a triangle, which minimizes the sum of the absolute distances to the edges, seems to be a different point then the Lemoine point.
How can this point be determined and why might the Lemoine point be preferable?
Edit: Since this question has been marked as a possible duplicate, based on the claim that this point and the Lemoine point are the same: They are not. If you consider the triangle spanned by the points (-2,0), (2,0), (0,2), the total distance to the edges of the point (0,2) is the height 2. This is smaller then the total distance of the Lemoine point to the edges, which is approx. 2.41.
The point with the minimum total distance always seems to come down to the vertex with the minimum height, but I am not shure how to prove it.
Edit2: After reconsideration, the answer by Calvin Lin in the referenced question, seems to answer my first question. However, I still wonder, in which cases it might be beneficial to consider the squared instead of the absolute distances.