Here is a problem that I've proposed which I believe to be true but I haven't verified it .

Take any triangle ABC and three concurrent lines emanating from each vertex A,B,C which extend to the sides x,y,z respectively .Prove that the area of triangle AZY + triangle ZBX + triangle YXC >=3(area of XZY). Equality is reached when the points x,y and z are the mid points of the sides of the triangle .