Q: Prove between Simson line & Nine point circle. Prove that the Simson lines of diametrically opposite points on the circumcircle are perpendicular to each other and meet on the nine-point circle.
I proved the first part of the problem but not able to prove second part, please provide the proof of second part.
For the prove of first part:
Lemma:
Angle between Simson line is half the angular measure of the arc between the point (on circumcircle).
Hence angular measure of the arc = $180$ so it directly follows from this lemma.

A: For the second part you can use the following lemmas: 
Lemma 1. Simson line of point $P$ bisects segment $PH$, where $H$ is the orthocenter of triangle $ABC$.
Lemma 2. Nine-point circle is the image of circumcircle in the homothety with center $H$ and scale $\frac 12$.
