Problem
GEO-B3-M05-P012 Four Euler Lines
#12
★★★☆☆ Level 3 of 5
With the notation of the previous problem, prove that the Euler lines of triangles \(ABC\), \(HBC\), \(AHC\), and \(ABH\) are concurrent.
Inspired by Prasolov special points geometry method
C. Hint 1. All four nine-point circles coincide.
D. Hint 2. The Euler line of each triangle passes through the center of its nine-point circle.
By the previous problem, the four triangles have the same nine-point circle. Let its center be \(N\).
In any triangle, the center of the nine-point circle lies on the Euler line: it is the midpoint between the circumcenter and the orthocenter of that triangle. Hence each of the four Euler lines passes through \(N\). Therefore they are concurrent at \(N\).
Best placed right after the common nine-point circle problem.