Question 3
In the plane, 16 pairwise distinct points are given, denoted $A_{i, j}$ for $i, j \in \{1,2,3,4\}$. These points satisfy the following collinearity and concyclicity relations:
$\triangleright$ for every $i \in \{1,2,3,4\}$, the points $A_{i, 1}, A_{i, 2}, A_{i, 3}$ and $A_{i, 4}$ are collinear;
$\triangleright$ for every $j \in \{1,2,3,4\}$, the points $A_{1, j}, A_{2, j}, A_{3, j}$ and $A_{4, j}$ are collinear;
$\triangleright$ the quadrilaterals $A_{1,1} A_{1,2} A_{2,2} A_{2,1}, A_{2,1} A_{2,2} A_{3,2} A_{3,1}, A_{3,1} A_{3,2} A_{4,2} A_{4,1}, A_{1,2} A_{1,3} A_{2,3} A_{2,2}$, $A_{1,3} A_{1,4} A_{2,4} A_{2,3}, A_{1,1} A_{2,2} A_{2,3} A_{1,4}$ and $A_{1,1} A_{2,2} A_{3,2} A_{4,1}$ are cyclic.
Prove that the quadrilateral $A_{4,1} A_{3,2} A_{3,3} A_{4,4}$ is also cyclic.
Note: a quadrilateral is called cyclic if its four vertices lie on one circle.
$\triangleright$ for every $i \in \{1,2,3,4\}$, the points $A_{i, 1}, A_{i, 2}, A_{i, 3}$ and $A_{i, 4}$ are collinear;
$\triangleright$ for every $j \in \{1,2,3,4\}$, the points $A_{1, j}, A_{2, j}, A_{3, j}$ and $A_{4, j}$ are collinear;
$\triangleright$ the quadrilaterals $A_{1,1} A_{1,2} A_{2,2} A_{2,1}, A_{2,1} A_{2,2} A_{3,2} A_{3,1}, A_{3,1} A_{3,2} A_{4,2} A_{4,1}, A_{1,2} A_{1,3} A_{2,3} A_{2,2}$, $A_{1,3} A_{1,4} A_{2,4} A_{2,3}, A_{1,1} A_{2,2} A_{2,3} A_{1,4}$ and $A_{1,1} A_{2,2} A_{3,2} A_{4,1}$ are cyclic.
Prove that the quadrilateral $A_{4,1} A_{3,2} A_{3,3} A_{4,4}$ is also cyclic.
Note: a quadrilateral is called cyclic if its four vertices lie on one circle.