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## On the figure of IMO 2009 P2 Generalization

For the generalization, see the file below:

IMO P2 generalization

Problem 1: Given two triangles $ABC$  and $XYZ$ with their circumcircle $(O)$. $YZ$ intersects $AB, AC$ at $P, N$, respectively. Let $J$ be the projection of $O$ on $YZ$. Let $K, L$ be the midpoints of $BN, CP. A'$ be the reflection of $J$ wrt $KL$. Similarly we define $B', C'$. Prove that:

1 (Nguyen Van Thanh).  $A'$ lies on the 9-point circle of triangle $ABC$.

2.  Two triangles $XYZ$ and $A'B'C'$ are similar.

Problem 2: Given a triangle $ABC$ and its circumcircle $(O)$. Let $d$ be an arbitrary line.  $d$ intersects $BC, CA, AB$ at $X, Y, Z$. Let $L$ be the projection of $O$ on $d$. Prove that $(ALX), (BLY), (CLZ)$ are coaxal.

Problem 3:  Given triangle $ABC$, its circumcircle $(O)$ and a vector $\vec{u}$ . Let $d$ be a line such that $d // \vec{u}$. $d$ intersects $AB, AC$ at $Y, Z$. Let $M, N$ be the midpoints of  $BY, CZ$, respectively; $L$ be the projection of $O$ on $d$. Prove that when $d$ moves, the line through $L$ and perpendicular to $MN$ passes through a fixed point.

Have fun! 🙂