Geometry · Euler line · Circumcentre and orthocentre · Isosceles triangle · Congruent triangles
The line through the circumcentre and the orthocentre of a triangle \(ABC\) (the Euler line of the triangle) crosses the interior of the side \(CA\) at a point \(M\) and the interior of the side \(CB\) at a point \(N\), and
\[ CM = CN . \]
Prove that \(\angle ACB = 60^\circ\).
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Serbian Regional Competition 2005, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source