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well known property of centroid with parallel line from A

Source: Azerbaijan NMO 2022, Senior P5

August 12, 2023
geometryCentroidcircumcircle

Problem Statement

Let ω\omega be the circumcircle of an acute angled tirangle ABC.ABC. The line tangent to ω\omega at AA intersects the line BCBC at the point T.T. Let the midpoint of segment ATAT be N,N, and the centroid of △ABC\triangle ABC be the point G.G. The other tangent line drawn from NN to ω\omega intersects ω\omega at the point L.L. The line LGLG meets ω\omega at S≠L.S\neq L. Prove that AS∥BC.AS\parallel BC.
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