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 3 of 5: Choose a cyclic triple of projections

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BC=3BU,\qquad U\in BC,\quad U\mapsto V\in CA\mapsto T\in AB

Detailed analysis

We may therefore assume that every side contains at least two points of each color. Choose $U\in BC$ with $BC=3BU$ . Let $V\in CA$ be the orthogonal projection of $U$ onto $CA$ , and let $T\in AB$ be the orthogonal projection of $V$ onto $AB$ . In an equilateral triangle, the projection of $T$ onto $BC$ is again $U$ ; hence $UV\perp CA$ , $VT\perp AB$ , and the three points form the standard cyclic configuration of the proof.