BG-MO 1964

There are given two intersecting lines $g_1,g_2$ and a point $P$ in their plane such that $\angle(g1,g2)\ne90^\circ$ . Its symmetrical points on any point $M$ in the same plane with respect to the given lines are $M_1$ and $M_2$ . Prove that:

(a) the locus of the point $M$ for which the points $M_1,M_2$ and $P$ lie on a common line is a circle $k$ passing through the intersection point of $g_1$ and $g_2$ . (b) the point $P$ is an orthocenter of a triangle, inscribed in the circle $k$ whose sides lie at the lines $g_1$ and $g_2$ .

Let $a_1,b_1,c_1$ are three lines each two of them are mutually crossed and aren't parallel to some plane. The lines $a_2,b_2,c_2$ intersect the lines $a_1,b_1,c_1$ at the points $a_2$ in $A$ , $C_2$ , $B_1$ ; $b_2$ in $C_1$ , $B$ , $A_2$ ; $c_2$ in $B_2$ , $A_1$ , $C$ respectively in such a way that $A$ is the perpendicular bisector of $B_1C_2$ , $B$ is the perpendicular bisector of $C_1A_2$ and $C$ is the perpendicular bisector of $A_1B_2$ . Prove that:

(a) $A$ is the perpendicular bisector of $B_2C_1$ , $B$ is the perpendicular bisector of $C_2A_1$ and $C$ is the perpendicular bisector of $A_2B_1$ ; (b) triangles $A_1B_1C_1$ and $A_2B_2C_2$ are the same.