Covering a triangle with positive and negative homothetic copies (Q2464361)

Covering of discs in the Euclidean plane is discussed in somewhat general situation and a particular problem is resolved. The author proves that if \(\Delta\) ia a triangle and \(\Delta_1=x_1\Delta,\Delta_2=x_2\Delta,\dots,\Delta_n=x_n\Delta\) is a set of homothetic copies of \(\Delta\), \(x_1 \geq x_2 \geq \cdots \geq x_n>0\) and \(\sum_{i=1}^n x_i^2 \geq 1+x_2\) then there exist signs \(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n \in \{\pm 1\}\) such that \(\varepsilon_1\Delta_1,\varepsilon_2\Delta_2,\dots,\varepsilon_n\Delta_n\) cover \(\Delta\). This is used for proving that if \(\Delta\) is an isosceles right triangle of area at most \((1+\sqrt{2})/2\) times the total area of \(\Delta_1,\Delta_2, \dots,\Delta_n\) then \(\Delta\) can be covered by rotations of \(\Delta_i\) by integer multiples of \(\pi/4\).