Iranian Geometry Olympiad 2020 (Elementary) P4
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Let $P$ be an arbitrary point in the interior of triangle $ABC$. Lines $BP$ and $CP$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$, respectively. Let the lines through $L$ and $K$ parallel to $CF$ and $BE$ intersect $BC$ at $S$ and $T$, respectively; moreover, denote by $M$ and $N$ the reflection of $S$ and $T$ over the points $L$ and $K$, respectively. Prove that as $P$ moves in the interior of triangle $ABC$, line $MN$ passes through a fixed point.
- Anindya Biswas
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Re: Iranian Geometry Olympiad 2020 (Elementary) P4
Here's a simple argument using congruence and similarity of triangles and properties of parallel lines:
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann