Problem

Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade

Tags: geometry



Given a parallelogram \(ABCD\). Let \(\ell_1\) be the line through \(D\), parallel to \(AC\), and \(\ell_2\) the external bisector of \(\angle ACD\). The lines \(\ell_1\) and \(\ell_2\) intersect at \(E\). The lines \(\ell_1\) and \(AB\) intersect at \(F\), and the line \(\ell_2\) intersects the internal bisector of \(\angle BAC\) at \(X\). The line \(BX\) intersects the circumcircle of triangle \(EFX\) at a second point \(Y\). The internal bisector of \(\angle ACD\) intersects the circumcircle of triangle \(ACX\) at a second point \(Z\). Prove that the quadrilateral \(DXYZ\) is inscribed in a circle.