29th Junior Balkan Mathematical Olympiad (JBMO 2025 )

Problem 3. Let ABC be a right-angled triangle with ∡BAC=90°, let D be the foot of the altitude from A to BC, and let E be the midpoint of DC. The circumcircle of ΔABD intersects AE again at point F. Let X be the intersection of the lines AB and DF. Prove that

\bar{X D} = \bar{X C}

Solution 1. Solution 1

Solution 2. Solution 2

Solution 3. Solution 3