Consider 2014 distinct parallels lines $l_1,..,l_{2014}$. Since any line contains infinitely many number of points, in each line there exist an infinitely many number of point with the same color. Since $2014>2013$ there exist $1\leq i<j\leq 2014$ such that the lines $l_i,l_j$ contain infinitely many points of the same color, say color $X$. Take two points on the line $l_j$ which has the color $X$. Now all the triangle formed by these two points and a point with color $X$ in the line $l_i$ have the color $X$ and the same area, QED