Can a median divide a triangle in equal area?

Answer 1

Sure. That's true of a median in every isosceles triangle,

and every median in an equilateral triangle.

In fact it is true for any median of any triangle. The two parts may not be the same shapes but they will have the same area. That is why the point where the three medians meet (centroid) is the centre of mass of a triangular lamina of uniform thickness.