XXXI Brazilian Math Olympiad

The tangent lines passing through $A$ are perpendicular to $AX$ and $AY$, so $AX\perp AC$ and $AY\perp AB$. Since $AXPY$ is a parallelogram, $PX$ is parallel to $AY$, so $PX\perp AB$. Since $AB$ is a chord of a circle with center $X$, $PX$ is the perpendicular bisector of $AB$. Analogously, $PY$ is the perpendicular bisector of $AC$. So $P$ is the intersection of the perpendicular bisectors of $AB$ and $AC$.

and, therefore, circumcenter of the triangle $ABC$.