aops. square Jean-Louis Ayme, https://artofproblemsolving.com/community/c6t48f6h2607184_a_circle_tangent_to_the_diagonal_of_a_square

 Let ABCD be a square, (A) the circle (A,AB), I midpoint of AB, segment CI intersects the circle (A) at Q. Prove that AC tangents the circle (BIQ).

Proof
Clearly CI passes through K, reflection of D about A, thus <IQB=45. Take O, midpoint of AC; as <BOI=45, QOIB is cyclic, OB is its circumdiameter, perpendicular to AC, done.