We prove that the interior of every convex polygon with n vertices (n ≥ 4) can be illuminated by four 45°-vertex lights. We restrict each vertex to anchoring at most one floodlight. This answers
a question of O’Rourke, Shermer and Streinu .
For a given triangle, we consider several sequences of nested triangles
obtained via iterative procedures. We are interested in the limiting behavior
of these sequences. We briefly mention the relevant known results and prove
that the triangle determined by the feet of the angle bisectors converges in
shape towards an equilateral one. This solves a problem raised by Trimble~.