Question 1
Let \(ABC\) be an acute scalene triangle. Let \(D,E,F\) be the midpoints of \(BC,CA,AB\), and let \(O,O'\) be its circumcenter and nine-point center. A point \(P\) lies inside triangle \(DEF\). Lines \(DP,EP,FP\) meet the nine-point circle again at \(D',E',F'\). Reflect \(A,B,C\) in \(D',E',F'\) to obtain \(A',B',C'\), respectively.
(a) If \(PO=PO'\), prove that the circle \((A'B'C')\) passes through \(O\).
(b) Reflect \(A',B',C'\) in lines \(OD,OE,OF\), obtaining \(X,Y,Z\). Let \(H\) be the orthocenter, and put \(M=HX\cap BC\), \(N=HY\cap CA\), \(K=HZ\cap AB\). Prove that \(M,N,K\) are collinear.
Editorial domain note: part (b), as printed, permits positions where its lines or intersections do not exist. The proof below concerns configurations in which the named intersections are finite and defined; a concrete exception is exhibited in the solution.
(a) If \(PO=PO'\), prove that the circle \((A'B'C')\) passes through \(O\).
(b) Reflect \(A',B',C'\) in lines \(OD,OE,OF\), obtaining \(X,Y,Z\). Let \(H\) be the orthocenter, and put \(M=HX\cap BC\), \(N=HY\cap CA\), \(K=HZ\cap AB\). Prove that \(M,N,K\) are collinear.
Editorial domain note: part (b), as printed, permits positions where its lines or intersections do not exist. The proof below concerns configurations in which the named intersections are finite and defined; a concrete exception is exhibited in the solution.