Question 14
Let $A_1A_2A_3A_4$ be a convex quadrilateral that is **not** cyclic and whose opposite sides are **not** parallel. For $1 \le i \le 4$, let $M_i$ be the midpoint of $A_{i-1}A_{i+1}$. Let $B_i$ be a point on the tangent to the circumcircle of triangle $A_{i-1}A_iA_{i+1}$ at $A_i$, such that the reflection of $M_i$ over the angle bisector of $\angle A_{i+1}A_{i+2}A_{i+3}$ lies on line $B_iA_{i+2}$. Let $C_i$ be the unique intersection point of lines $A_iA_{i+1}$ and $B_iB_{i+1}$. (All indices are modulo 4.)
Prove that $C_1, C_2, C_3, C_4$ are collinear.
Prove that $C_1, C_2, C_3, C_4$ are collinear.