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← Drzavno takmicenje (National round) 2026National

Geometry · Circumcircle · Orthocentre · Centroid · Euler line · Central symmetry · Construction

Problem 2, 2026

NationalProof

Let \(ABC\) be a triangle and let \(k\) be its circumcircle, with centre \(O\). Construct a point \(D\) on \(k\) such that the centroids of the triangles \(ABC\) and \(ABD\) are collinear with the point \(O\).

A B C O T1
The data: \(ABC\) inscribed in \(k\), the centre \(O\) and the centroid \(T_1\) of \(ABC\). The point \(D\) must be placed on \(k\) so that the centroid of \(ABD\) falls on the dashed line \(T_1O\).

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Serbian National Competition (Drzavno takmicenje) 2026, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source

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