Let $P$ be a point in a closed continuous curve in the plane that's non self-intersecting. Show we can find two points on the curve whose midpoint is P.
I found the solution below, but I have some questions regarding it:
- Why must m lie inside or on C'? I tried obtaining a contradiction by assuming otherwise, but I'm not sure how.
- Similarly, why must M lie outside or on C'?
- Why must P be the midpoint of QQ'? Is it because the points are collinear and $PQ= PQ'$?
Imagine if $m$ was outside of $C'$. Then $Pm$ would intersect $C'$ at some point $m'$ closer to $P$ than $m$. Rotate $m'$ by $180^\circ$ around $P$ and you would get a point $m''$ on $C$ such that $Pm''=Pm'\lt Pm$, which contradicts the choice of $m$ as the "closest" point.
is very similar to 1, try it as an exercise.
Rotation of $180^\circ$ around $P$ maps $C$ to $C'$ and $C'$ to $C$. Thus, if you have a point $Q$ on both $C$ and $C'$, its rotated image $Q'$ will be on both the rotated images of $C$ and $C'$, that is, it will be on both $C'$ and $C$. Now $PQ=PQ'$ simply because that is how the rotation works - the distance from the centre $P$ stays unchanged. Also the angle $\angle QPQ'$ is a straight angle (as this is the angle of rotation - $180^\circ$). Thus $P$ is the midpoint of $QQ'$ as you said.