The centroid - where the medians meet.
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Center of mass of an equilateral triangle is located at its geometric center (centroid).
it would come down to the type of triangle.
Yes and no. Each median divides the triangle into two such that for any point on the median, the mass on one side is balanced by the mass on the other. But the mass ahead of that point may or may not balance the mass behind. It is the point of intersection of the medians - the centroid - which is the centre of mass or centre of balance of the triangle.
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The center of gravity of a triangle is its centroid. The centroid of a triangle is the intersection of the three medians.