Question 84
In an acute triangle $ABC$ we have $|AB| \gt |BC|$, and the points $A_1$ and $C_1$ are the feet of altitudes from the vertices $A$ and $C$, respectively. Let $D$ be the second intersection of the circumcircles of triangles $ABC$ and $A_1BC_1$ (different from $B$). Let $Z$ be the intersection of the tangents of the circumcircle of the triangle $ABC$ at points $A$ and $C$, and let lines $ZA$ and $A_1C_1$ meet at the point $X$, and lines $ZC$ and $A_1C_1$ at the point $Y$.
Prove that the point $D$ lies on the circumcircle of the triangle $XYZ$.
Prove that the point $D$ lies on the circumcircle of the triangle $XYZ$.