Question 2
In the acute triangle $ABC$, $M$ is a point in the interior of segment $AC$ and $N$ is a point on the extension of segment $AC$ such that $MN=AC$. Let $D,E$ be the feet of the perpendiculars from $M,N$ onto the lines $BC,AB$, respectively. Prove that the orthocentre of $\triangle ABC$ lies on the circumcircle of $\triangle BED$.
Editorial qualification: the source figure places $N$ beyond $C$. Even with that interpretation, the hypotheses can allow $E=B$, so the stated circumcircle is then undefined. The original statement is preserved; the proof below assumes $N$ beyond $C$ and $B,E,D$ distinct. This question requires review.
Editorial qualification: the source figure places $N$ beyond $C$. Even with that interpretation, the hypotheses can allow $E=B$, so the stated circumcircle is then undefined. The original statement is preserved; the proof below assumes $N$ beyond $C$ and $B,E,D$ distinct. This question requires review.