76-A generalization of Simson line theorem

Problem 1: Let $ABC$ be a triangle, let a line $(L)$ through circumecenter and a point $P$ lie on circumcircle.

Let $AP,BP,CP$ meets $(L)$ at $A_P, B_P, C_P$. 

Denote $A_0,B_0,C_0$ are projection (mean perpendicular foot) of $A_P, B_P, C_P$ to $BC,CA,AB$ respectively.

Then $A_0,B_0,C_0$ are collinear.

- When $(L)$ through $P$, this line is Simson line.

Problem 2: The new line $\overline {A_0B_0C_0}$ bisect the orthocenter and P