More general: $ABCD$ and $A_1B_1C_1D_1$ are any squares. Prove that there exists a permutation of the vertices of $ABCD$ such that $AA_1$, $BB_1$, $CC_1$, and $DD_1$ are concurrent.

What is $A_1$?

I think it is wrong.it must be $ABCD$ and $A_1B_1C_1D_1$

No I think both squares contain the point $A$

This problem is the homogeneity theorem.