12.4. In space, there are two skew lines and a point $A$ on one of them. Through these lines, two perpendicular planes are drawn, forming a right dihedral angle. Find the geometric locus of the projections of point $A$ on the edges of such angles.
12.4. Let $\pi_{1}$ and $\pi_{2}$ be perpendicular planes passing through lines $l_{1}$ and $l_{2}$; $l$ is the line of their intersection; $X$ is the projection of point $A$, lying on line $l_{1}$, onto line $l$. Draw

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-224.jpg?height=319&width=436&top_left_y=226&top_left_x=572)

Fig. 86 through point $A$ a plane $\Pi$ perpendicular to line $l_{2}$. Since $\Pi \perp l_{2}$, then $\Pi \perp \pi_{2}$. Therefore, line $A X$ lies in plane $\Pi$. Hence, if $B$ is the point of intersection of $\Pi$ and $l_{2}$, then $\angle B X A=90^{\circ}$, i.e., point $X$ lies on the circle with diameter $A B$ constructed in plane $\Pi$.