MathLabs

Problem 4

Let ABCABC be an equilateral triangle and let E\mathcal E be the union of its three sides. Determine whether every partition of E\mathcal E into two disjoint subsets has one subset containing the vertices of a right-angled triangle. Justify your answer.
Step 2 of 5: The easy case
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some side has at most one blue point⇒the other two sides force a blue right triangle\text{some side has at most one blue point}\Rightarrow\text{the other two sides force a blue right triangle}
Detailed analysis

Suppose one side contains at most one blue point. Apply the projection lemma to pairs of red points on that side and the corresponding points on the other two sides. It forces the relevant points on those sides to be blue; among them the equilateral-triangle geometry supplies three blue points forming a right angle, a contradiction.