[ Properties of sections ]

On a plane, there are three rays with a common origin. They divide the plane into three obtuse angles, and a point is taken inside each of these angles. Using a compass and a straightedge, construct a triangle whose vertices lie on the given rays, and whose sides pass through the given points.

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Let $O X, O Y, O Z$ be given rays located in the same plane, and points $A, B$, and $C$ lie inside the angles $X O Y$, $X O Z$, and $Y O Z$ respectively. We will consider that we have a planar image of a certain trihedral angle, and points $A, B$, and $C$ are images of points lying in its faces. We will construct the image of the section of this trihedral angle by a plane passing through points $A, B$, and $C$. On the ray $O Z$, take an arbitrary point $M$. Let the line $M B$ intersect the ray $O X$ at point $K$, and the line $M C$ intersect the ray $O Y$ at point $N$. Then the lines $K N$ and $B C$ lie in the same plane. Suppose they intersect at point $L$. Then the line $A L$ lies in the plane $X O Y$ and in the plane of the desired section. If $P$ and $Q$ are the points of intersection of the line $A L$ with the rays $O X$ and $O Y$ respectively, then the lines $P B$ and $Q C$ are either parallel (in which case the further construction is obvious), or they intersect at a point $R$ lying on the ray $O Z$. In the latter case, the triangle $P Q R$ is the desired one.