It is the line from a vertex of the triangle to the midpoint of the side opposite that vertex.
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Yes in equilateral triangle.
An isosceles or an equilateral triangle perhaps?
Yes, if the triangle is isosceles or equilateral.
In an isosceles or equilateral triangle, when from the vertex that is different from the others.
Yes. In an isosceles or equilateral triangle, it always is.