When the two perspectivities are not co-planar, and the lines bearing our three equivalent intervals are all taken though a single point, as above, we obtain the famous Desargues configuration, which concerns a particular relationship between two triangles, here Δ TUV and Δ T'U'V'.
The truth of Desargues Theorem is trivially obvious if the triangles
Δ TUV and
Desargues: the deep base → Five Planes
However, the illustrations have two triangles actually in the same plane - that of the screen: it is a set of ‘shadows’ (projections), from a point (‘centre’), of spatially distributed elements. This coplanar case is the degenerate case: given a coplanar case such as this, we are always be able to (re)generate a spatial, non-degenerate construction, and a centre, that together give rise to it.
In fact, a “normal” eye always does this quite involuntarily (consider that a one-eyed person has a sense of depth, without seeing in 3D). As an artist, I do this in reverse, so to say: I imagine a 3D scene, and “flatten” it on to the drawing surface – as above. Whether or not I do so consciously, I use Desargues.