Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Prove it Belarus

The diagonals of a quadrilateral $ABCD$ intersect at the point $N$ . It is known that the incircles of the triangles $ABN$ and $CBN$ are tangent to each other. Moreover, the incircles of the triangles $ADN$ and $CDN$ are tangent to each other as well.
Find all possible values of $AN : NC$ .

Solution

$AN : NC = 1$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.