MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 added 51 characters in body

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 $a_4=1$.
1

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 $a_4=1$.