1.32. On three mutually perpendicular lines intersecting at point $O$, points $A, B$, and $C$ are given, equally distant from $O$. Let $l$ be an arbitrary line passing through $O$; points $A_{1}, B_{1}$, and $C_{1}$ are symmetric to $A, B$, and $C$ with respect to $l$. Planes passing through points $A_{1}, B_{1}$, and $C_{1}$ perpendicular to lines $O A, O B$, and $O C$ respectively, intersect at point $M$. Find the geometric locus of points $M$.

## Problems for Independent Solving
1.32. Let's direct the coordinate axes along the rays $O A, O B$ and $O C$. Suppose the line $l$ forms angles $\alpha, \beta$ and $\gamma$ with these axes, respectively. The coordinates of point $M$ are equal to the coordinates of the projections of points $A_{1}, B_{1}$ and $C_{1}$ onto the axes $O x, O y$ and $O z$ respectively, i.e., they are equal to $a \cos 2 \alpha, a \cos 2 \beta$ and $a \cos 2 \gamma$, where $a=|O A|$. Since $\cos 2 \alpha + \cos 2 \beta + \cos 2 \gamma = 2\left(\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma\right) - 3 = -1$ (see problem 1.21) and $-1 \leqslant \cos 2 \alpha, \cos 2 \beta, \cos 2 \gamma \leqslant 1$, the desired locus of points consists of the points of intersection of the cube $|x|,|y|,|z| \leqslant a$ with the plane $x+y+z=-a$; this plane passes through the vertices with coordinates $(a,-a,-a):(-a, a,-a)$ and ( $-a,-a, a)$.