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Give collineations to prove the following (in the extended projective plane):

a, One cannot contruct the midpoint of two points using a straightedge only.

b, One cannot construct the reflection of a line to a parallel line using a straightedge only.

So the task is to find collineations (given by coordinate transformations) where the image of the midpoint (reflection) is not the midpoint (reflection) of the image.

(You can only construct something with a straightedge only, if for all collineations, the images of the objects (lines, points) satisfy the requirements (e.g. the midpoint of the image points is the image of the midpoint))

Nesa
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1 Answers1

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Hint: Euclidean concepts like midpoint depend on the points at infinity having a special role. Therefore you should concentrate on collineations which break that role, i.e. which turn parallel lines into non-parallel ones. So I'd start by picking two points $A$ and $B$, finding the line $g$ connecting these and describing another line $h$ parallel to it. Then find a collineation which fixes $A$ and $B$ but moves $h$ so that it intersects $g$ in a finite point.

You can use this post to find a description for such a transformation in terms of coordinates if you can't think of an easier way to come up with a particularly simple counterexample. Or you can mix things, and see which points would make the computation from that post particularly simple.

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MvG
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