Consider $2014$ parallel lines. Each line contains infinitely many points. Since the number of the colors is finite, by the pigeonhole principle, there exist on each line infinitely many points of the same color. Choose for each line one color for which there exist infinitely many points. Since there are $2013$ colors and $2014$ lines, by the pigeonhole principle there exist at least two lines $\ell_1$, $\ell_2$ for which the same color $\mathcal{C}$ has been chosen. Choose two points $B$, $C$ from the first line $\ell_1$ of this color $\mathcal{C}$. Choose infinitely many points $A_1$, $A_2$, $\ldots$ from the second line $\ell_2$ of this color $\mathcal{C}$. Triangles $A_1BC$, $A_2BC$, $\ldots$ are all of the same color $\mathcal{C}$ and have the same area since $\ell_1$, $\ell_2$ are parallel.