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← Okruzno takmicenje (Regional round) 2026Regional

Geometry · Vectors · Central symmetry · Centroid

Problem 2, 2026

RegionalProof

Let \(ABC\) be a triangle and let \(X\) be a point of its plane. Denote by \(A'\), \(B'\), \(C'\) the images of \(A\), \(B\), \(C\) under the reflection in the point \(X\). Let \(M\), \(N\), \(P\) be the midpoints of the segments \(AB'\), \(BC'\), \(CA'\) respectively.

Prove that \(X\) is the centroid of the triangle \(MNP\).

A B C A' B' C' X M N P
The configuration: \(X\) halves each of \(AA'\), \(BB'\), \(CC'\), and \(M\), \(N\), \(P\) halve the three segments drawn across it.

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

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