Geometry · Orthocentre · Cevians · Cyclic quadrilaterals · Circumcentre
Problem 2, 2023
On the sides of an acute triangle \(ABC\) points \(A_1 \in BC\), \(B_1 \in CA\) and \(C_1 \in AB\) are chosen so that
\[ \angle CC_1B = \angle AA_1C = \angle BB_1A = \varphi, \]
where \(\varphi\) is an acute angle. The segments \(AA_1\), \(BB_1\) and \(CC_1\) meet one another in the points \(M\), \(N\) and \(P\). Prove that the orthocentre of the triangle \(ABC\) is the centre of the circle circumscribed about the triangle \(MNP\).
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Serbian Regional Competition (Okruzno takmicenje) 2023, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source