Problem

Source: 2022 USAJMO 4

Tags: geometry, coordinate geometry, USAJMO, USA, geometry solved, AMC, USA(J)MO



Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus. Proposed by Ankan Bhattacharya