Problem
GEO-B2-M01-P106 Orthocenter and a Diameter Circle
In an acute triangle \(ABC\), the orthocenter is \(H\). Points \(D\) and \(E\) are the feet of the altitudes from \(B\) and \(C\), respectively. It is known that \(\angle BAC=85^\circ\). Prove that \(A,D,H,E\) lie on one circle, and find \(\angle DHE\).
C. Hint 1. Find two right angles standing on the same segment.
D. Hint 2. Introduce the circle with diameter \(AH\) and compare angles subtending chord \(DE\).
E. Full solution. Since \(D\) is the foot of the altitude from \(B\), we have \(BD\perp AC\). Point \(H\) lies on this altitude, and \(A,D,C\) are collinear; hence \(DH\perp AD\), so \(\angle ADH=90^\circ\).
Similarly, \(E\) is the foot of the altitude from \(C\), so \(EH\perp AE\) and \(\angle AEH=90^\circ\).
Thus both \(D\) and \(E\) see segment \(AH\) under a right angle. Therefore \(D\) and \(E\) lie on the circle with diameter \(AH\). Hence \(A,D,H,E\) are concyclic.
In this circle, angles \(\angle DHE\) and \(\angle DAE\) subtend the same chord \(DE\). Therefore \(\angle DHE=\angle DAE=\angle BAC=85^\circ\).
Method comment. points \(D\) and \(E\) see segment \(AH\) under a right angle, so they should be placed on one circle 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 6: a medium regional-style problem. A direct angle chase quickly overloads the diagram, so one must see a hidden circle or replace an angle by a tangent argument; after that, the chain is short.
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.