You can drag •points P and S, •the end-points of the black interval, and •most of the lines.
Here we have two perspectivities, centred on P and S, comparing a black interval on its own black line with two intervals, one red, one blue, each on a line with the same colour as itself.
By the perpectivities,
It follows that the red interval is identical or equivalent to the blue, as both are equivalent to the same thing.
To paraphrase Euclid's ‘First Common Notion’ [video] —
“Things that are
equivalent to the same thing
A projective comparison,
such as this,