Three trivial observations (not an answer).
Let us denote by $a_k(\Delta)$ the maximal covered area in $\Delta$. Then
- for any right triangle $a_2=1$,
- for any acute triangle $a_3=1$,
- for any triangle $a_4=1$.
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Let us denote by $a_k(\Delta)$ the maximal covered area in $\Delta$. Then
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