Problem

Source: Czech-Polish-Slovak Junior Match 2018, Team p2 CPSJ

Tags: ratio, areas, geometry, symmetry, right triangle



Given a right triangle $ABC$ with the hypotenuse $AB$. Let $K$ be any interior point of triangle $ABC$ and points $L, M$ are symmetric of point $K$ wrt lines $BC, AC$ respectively. Specify all possible values for $S_{ABLM} / S_{ABC}$, where $S_{XY ... Z}$ indicates the area of the polygon $XY...Z$ .