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Olimpiade Italiana di Matematica

Geometry Difficulty 4.6 AIME Prove it Italy

Problem:

Let $A B C D E F$ be a hexagon inscribed in a circle such that $A B = B C$ , $C D = D E$ and $E F = A F$ . Prove that the segments $A D$ , $B E$ and $C F$ are concurrent (that is, they have a point in common).

Solution

Solution:

Since $A B = B C$ , using the fact that in a circle congruent chords correspond to congruent inscribed angles, we have $\angle A E B = \angle B E C$ . Similarly $\angle C A D = \angle D A E$ and $\angle A C F = \angle F C E$ .

Hence $A D$ , $E B$ and $C F$ are the three bisectors of triangle $A C E$ and therefore they are concurrent at its incenter.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.