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Problem 880

AIME late

Geometry Difficulty 5.7 Prove it

$\left[\begin{array}{l}\text { Convex hull and supporting lines (planes). } \\ {[\underline{\text { Inequalities with angles }}]}\end{array}\right]$

Let there be five points in general position on a plane, that is, no three of them lie on the same line and no four lie on the same circle. Prove that among these points, there are two such that they lie on opposite sides of the circle passing through the remaining three points.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Among the given points, let's choose points $A$ and $B$ such that all other points lie on one side of the line $A B$ . The remaining three points will be denoted as $C, D, E$ such that $\angle A C B > \angle A D B > \angle A E B$ . Then points $C$ and $E$ lie on opposite sides of the circle passing through points $A, B$ , and $D$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.