Colling theorem: Let a line $(L)$ through the orthocener of the triangle ABC, reflection of $(L)$ in three sidelines concurrent be a point on the circumcircle(Colling point of $(L)$
Nice problem: Let $ABC$ be a triangle, let a line $(L)$ through the orthocenter of the triangle $ABC$ and $(L)$ cuts $AB, AC$ at $P, Q$. Denote $K$ be the intersection of two lines perpendicular to $AB,AC$ at $P,Q$. These line cut $BC$ at $M,N$. The circle through $KMN$ tangent the circumcircle at colling point of (L).
Note: $X_3$ of $ABC$ is $X_4$ of median triangle
Nice problem: Let $ABC$ be a triangle, let a line $(L)$ through the orthocenter of the triangle $ABC$ and $(L)$ cuts $AB, AC$ at $P, Q$. Denote $K$ be the intersection of two lines perpendicular to $AB,AC$ at $P,Q$. These line cut $BC$ at $M,N$. The circle through $KMN$ tangent the circumcircle at colling point of (L).
Note: $X_3$ of $ABC$ is $X_4$ of median triangle
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