Perpendicular Bisectors and Point Distributions

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Perpendicular Bisectors and Point Distributions

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Source: IDMC 2025 P3

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#1 h Apr 20, 2025, 5:03 AM

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Given $n\geq 6$ points in the plane such that all pairwise distances between them are distinct, prove that there exist two points $A$ , $B$ among them such that the perpendicular bisector of segment $AB$ has at least two points on each of its sides (i.e., each side contains at least one point other than $A$ and $B$ ).

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Quantum-Phantom

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Quantum-Phantom

#2 h Apr 20, 2025, 9:06 AM

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Call a set of points that satisfies the condition good. If the conclusion also holds, call the set nice.

Claim. The set consisting of the vertices of a convex $k$ -gon ( $k\ge5$ ) as well as an additional point is nice, if good.
Proof. Let $X$ be that additional point, and $V_1V_2\cdots V_k$ be that convex $k$ -gon such that $V_1X=\min_{1\le i\le k}\{V_iX\}$ . Since
$\angle V_2V_1V_3+\angle V_3V_1V_4+\cdots+\angle V_{k-1}V_1V_k=\angle V_2V_1V_k<\pi,$ then there exists $\angle V_iV_1V_{i+1}<\tfrac\pi{k-2}\le\tfrac\pi3$ . Hence, $\angle V_iV_1V_{i+1}<\max\{\angle V_iV_{i+1}V_1,\angle V_{i+1}V_iV_1\}$ , so $V_iV_{i+1}<\max\{V_iV_1,V_{i+1}V_1\}$ . WLOG let $V_iV_{i+1}<V_iV_1$ , then consider $\ell$ , the mediator of $V_{i+1}V_1$ . We see that $V_i$ is on the same side as $V_{i+1}$ and $X$ is on the same side as $V-1$ , of $\ell$ . Hence the set is nice.

Back to the original problem. Let the convex hull of the $n$ points be a convex $t$ -gon. If $t\le n-2$ , then there exists $2$ points inside the convex hell. Considering the mediator of these $2$ clearly works (otherwise the sole point must be outside the convex hull). If $t=n-1$ , then from $n-1\ge5$ , using the claim finishes. If $t=n$ , exclude a point from the convex hull, the remaining $n-1$ points still form a convex polygon, so using the claim finishes.

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