Question 116
Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^\circ$ and let $E$ be the foot of the perpendicular from $A$ on $BC$. Let $Z \neq A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T, E, B, Z$ are concyclic.