KöMaL 2025 A906

Let $\mathcal{V}_c$ denote the infinite parallel ruler with the parallel edges being at distance $c$ from each other. The following construction steps are allowed using ruler $\mathcal{V}_c$: $\bullet$ the line through two given points; $\bullet$ line $l'$ parallel to a given line $l$ at distance $c$ (there are two such lines, both of which can be constructed using this step); $\bullet$ for given points $A$ and $B$ with $|AB|\ge c$ two parallel lines at distance $c$ such that one of them passes through $A$, and the other one passes through $B$ (if $|AB|>c$, there exists two such pairs of parallel lines, and both can be constructed using this step). On the perimeter of a circular piece of paper three points are given that form a scalene triangle. Let $n$ be a given positive integer. Prove that based on the three points and $n$ there exists $C>0$ such that for any $0<c\le C$ it is possible to construct $n$ points using only $\mathcal{V}_c$ on one of the excircles of the triangle. We are not allowed to draw anything outside our circular paper. We can construct on the boundary of the paper; it is allowed to take the intersection point of a line with the boundary of the paper.