Problem 1.
Let D be any point on a side of triangle ABC. Construct a line through D that divides the triangle into two equal areas. When D corresponds to a vertex or to a midpoint, then a line along the median will suffice. Assume D does not necessarily correspond to a vertex or a midpoint.
Problem 2.
Construct the triangle with its extended sides and let the constucted line through D cut off, with lines AB and BC, a triangle half the area of the original regardless of whether D is between B and C.
Trace the midpoint of the segment cut off by the two sides (perhaps extended) of the triangle.
Or, trace an envelope of the lines through D as D is animated along BC.
