USAJMO 2022 Question 4

Let $ABCD$ be a rhombus. Points $K,L$ satisfy that $K$ is inside the rhombus, $L$ is outside it, and $KA=KB=LC=LD$. Prove that there are points $X,Y$ on lines $AC,BD$, respectively, such that $KXLY$ is a rhombus.

Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party compilation; https://aops.com/community/p24774741. One editorial point; manual grading.