Talk:Perpendicular from Point to Straight Line in Plane is Unique/Proof 1

Apart from an overcomplicated setup, what's different between this and Straight Line Perpendicular to Plane from Point is Unique? --prime mover (talk) 17:49, 12 November 2023 (UTC)

The second plane. Just having a straight line in the plane that intersects with the perpendicular isn't enough. --Telliott99 (talk) 12:45, 13 November 2023 (UTC)

Surely it is? Once you have specified $L$ and a point perpendicular to $L$ not in $\PP$ there is indeed only $1$ perpendicular. --prime mover (talk) 12:56, 13 November 2023 (UTC)

In fact I will go further and state that if such basic theorem cannot be found in the literature, it probably means it's trivially unnecessary. So here's a challenge: find this pointless theorem in print somewhere. --prime mover (talk) 12:58, 13 November 2023 (UTC)

I was missing something. L in plane P, point B on L. Thm says given point A in space, only one perp from A to P. Therefore, if A is perp to L at B and also perp to P, it's unique. But there are many other perps to L in space. So that doesn't help us without specifying the plane containing L and A. That plane must be perp to P and then it works.

And you're saying, start with A perp to P and also perp to L at B. OK, so you've done the additional work in a different way.

I will look through early Euclid again, but I'm pretty sure this, which is related to Axiom:Playfair's Axiom, is not spelled out. OTOH, I'm happy with proof 2 and its relatives. Simple, uses only Euclid I/7.