Problem
GEO-B2-M01-P082 Reflections of the Orthocenter
In an acute triangle \(ABC\), point \(H\) is the orthocenter. Points \(H_b\) and \(H_c\) are the reflections of \(H\) across lines \(AB\) and \(AC\), respectively. Prove that \(H_b\) and \(H_c\) lie on the circumcircle of triangle \(ABC\), and that quadrilateral \(B,C,H_c,H_b\) is cyclic.
C. Hint 1. Recall the value of angle \(\angle AHB\).
D. Hint 2. Reflection across \(AB\) sends angle \(\angle AHB\) to angle \(\angle AH_bB\).
E. Full solution. For the orthocenter of an acute triangle, \(\angle AHB=180^\circ-\angle ACB\).
Under reflection across line \(AB\), points \(A\) and \(B\) remain fixed, and \(H\) maps to \(H_b\). Hence \(\angle AH_bB=\angle AHB=180^\circ-\angle ACB\).
Thus the opposite angles of quadrilateral \(A,B,C,H_b\) sum to \(180^\circ\). Therefore \(A,B,C,H_b\) are concyclic, so \(H_b\) lies on the circumcircle of triangle \(ABC\).
In the same way, reflecting \(H\) across \(AC\), we obtain \(H_c\) on the same circumcircle.
Since \(B,C,H_b,H_c\) lie on one circle, namely the circumcircle of triangle \(ABC\), quadrilateral \(B,C,H_c,H_b\) is cyclic.
Method comment. reflection preserves the angle with the mirror line, and the angle at the orthocenter supplements an angle of the triangle Thus the solution is not a brute-force chase of all angles in the diagram, but a deliberate choice of the right circle or tangent, after which the angles can be compared through the same chord or the same line.
If the auxiliary step is skipped, the problem looks almost arbitrary: the equal angles live in different parts of the diagram. That is why the hidden configuration is identified first, then the angle replacement is made, and only at the end the required conclusion follows.
F. Difficulty justification. This is Level 7: a strong regional-style problem. It contains not only equal angles, but also a choice of an auxiliary configuration: a circle, a Miquel point, an orthic configuration, or a tangent. Without that first choice, the solution does not start.
G. Check. This is not a one-step exercise: it requires 3 key ideas. First one must recognise the hidden geometric structure, then make an angle replacement or add a circle, and only after that complete the final conclusion.