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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,
The red and blue intervals
are both projectively identical or equivalent
to the black interval,

because they are
projectively indistinguishable from it.

It follows that the red interval is identical or equivalent to the blue, as both are equivalent to the same thing.

To paraphrase the ‘First Common Notion’ [video]

“Two things that are equivalent to a third thing
are equivalent to each other.”

A projective comparison, such as this,
of two intervals via a third,

is a projectivity