Triangle ParentingGeometry Level 5
Let \(\bigtriangleup_1, \bigtriangleup_2, \ldots, \bigtriangleup_n\) be similar triangles (either directly or not) on a plane. The triangle \(\bigtriangleup_i\) has vertices named \(A_i\), \(B_i\), \(C_i\) at random. We call these parent triangles.
Every pair of parent triangles \(\left(\bigtriangleup_x,\bigtriangleup_y\right)\) has exactly one child triangle, which has vertices on the middle points of the segments \(A_xA_y\), \(B_xB_y\), \(C_xC_y\) connecting its parents.
No pair of parents has a degenerate child triangle.
Find the least \(n\) such that, for every choice of the initial triangles, there exist at least one child triangle similar to its parents.