What is the smallest equilateral triangle that can be guaranteed to cover any triangle whose longest side has length ? We might be tempted to think an equilateral triangle whose side length is also will do the job, as in (1) below.
However if a base angle is just over as shown in (2) the side marked is still the longest side but now the triangle we want to cover cannot fit into the equilateral triangle. Moving the equilateral triangle cannot help; the only way to cover two points that are apart is if they lie at vertices of the equilateral triangle and the same problem will arise whatever pair we pick.
Can you work out how large the equilateral triangle has to be before we can be confident it can cover any triangle whose longest side is ?