Let $A$ and $B$ be the subsets of the plane representing the colouring. Let $\alpha > 1$ be a real number and consider a ray from the origin $O$. If there are two points on this ray, $(X,Y) \in A \times B$ such that $\frac{|OY|}{|OX|}=\alpha$ then call this ray $(A,B)$-good. Similarly define $(B,A)$-good, $(A,A)$-good, and $(B,B)$-good. Any ray from $O$ has to be at least one of the four types of rays and possible multiple. As there are infinitely many rays from $O$ we can find three rays that are all, for example, $(A,B)$-good. Then there are six point $x_1,x_2,x_3 \in A$ and $y_1,y_2,y_3 \in B$ such that $\frac{|Oy_j|}{|Ox_j|}=\alpha$. These are the vertices of two similar monochromatic triangles with ratio $\alpha$.