Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an equilateral triangle and
![\mathcal{E}](/media/m/9/6/f/96f9aec2cdb69dfac9de0a898f10bd81.png)
the set of all points contained in the three segments
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
,
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
, and
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
(including
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
, and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
). Determine whether, for every partition of
![\mathcal{E}](/media/m/9/6/f/96f9aec2cdb69dfac9de0a898f10bd81.png)
into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.
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Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.