Geometry · Triangle centers · Circumcenter · Incenter · Orthocenter · Point reflection · Isogonal lines
In a triangle \(ABC\), let \(O\) and \(I\) be the centres of the circumscribed and of the inscribed circle, respectively. Let \(O_1\) be the image of \(O\) under the reflection in the point \(I\), so that \(I\) is the midpoint of the segment \(OO_1\), and let \(I_1\) be the image of \(I\) under the reflection in the point \(O\), so that \(O\) is the midpoint of the segment \(II_1\).
Suppose that \(O_1\) lies on the altitude from the vertex \(A\), and that \(I_1\) lies on the altitude from the vertex \(B\). Prove that the triangle \(ABC\) is equilateral.
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Serbian National Competition (Drzavno takmicenje) 2019, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source