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← Opštinsko takmičenje (Municipal round) 2019City

Geometry · Orthocenter · Altitudes · Concyclic points · Midlines

Problem 1, 2019

CityProof

Let \(AA_0\), \(BB_0\) and \(CC_0\) be the altitudes of a triangle \(ABC\), and let \(H\) be its orthocenter. Let \(M\) be the midpoint of the segment \(AA_0\) and let \(N\) be the midpoint of the segment \(CC_0\). Prove that the points \(M\), \(N\), \(B_0\) and \(H\) lie on one circle.

A B C A0 B0 C0 H M N
The three altitudes meet at \(H\); the highlighted points \(M\) and \(N\) are the midpoints of \(AA_0\) and \(CC_0\).

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Serbian Municipal Competition 2019, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source

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