Problem 4

Let

ABC

be an equilateral triangle and let

\mathcal E

be the union of its three sides. Determine whether every partition of

\mathcal E

into two disjoint subsets has one subset containing the vertices of a right-angled triangle. Justify your answer.

Step 4 of 5: Force three different colors

Previous step Next step

U,V,T\text{ cannot contain two points of the same color}

Detailed analysis

Apply the projection lemma on each of the three sides. If two of $U,V,T$ had the same color, the second point of that color guaranteed on the relevant side would force the third point to have the opposite color, and the cyclic relations would force the remaining point to have both colors. Thus $U,V,T$ must be pairwise differently colored.