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I ran into the sentence

Die Punkte A, B und C liegen auf einer Geraden.

From the dictionary I understand that "Gerade" takes on the form "Geraden" for both plural and genitive. But here it looks like the dative case, and it cannot be a masculine N-Noun (like Herr, Journalist, Student, etc.) because it's feminine. What's the explanation then?

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As you noticed, “auf einer Geraden” is dative, singular. The declination has the “n” in this case, because “Gerade” is a nominalisation of the adjective “gerade”. Compare:

Ich schenkte der Schönen ein Lächeln.

Hence your example is correct:

Die Punkte A, B und C liegen auf einer Geraden.

Another example:

Auf der Gegengeraden hat er das Tempo angezogen.

Also

Die Punkte A, B und C liegen auf einer Geodätischen.

(nominalisation), but

Die Punkte A, B und C liegen auf einer Geodäte.

(not a nominalisation in the technical sense).

However, apparently the origin of “Gerade” as a nominalisation has been somewhat lost, and the forms without the “n” are also used and deemed correct.

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It actually is Dativ:

The question to ask for 'Geraden' is "auf wem oder was?" which is the question indicating Dativ.

It should actually be "Auf einer Gerade", I agree to that and think you can use it like that, maybe 'Geraden' is even wrong on that place. Sadly us German-speakers aren't too fond with our own language at times. I think I found why it is used the way we see here:

Duden defines it as a 'substantiviertes Adjektiv', meaning its usage is derived from the adjective 'gerade' (straight). If you add 'Straße'(road) it becomes "Auf einer geraden Straße", which is correct.

I think that coudld be why it is used incorrectly, yet doesn't seem so.

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