Second Round, March, 2019

Geometry Difficulty 5.5 AIME, harder Prove it Netherlands

In a square $ABCD$ of side length $2$ we draw lines from each vertex to the midpoints of the two opposite sides. For example, we connect $A$ to the midpoint of $BC$ and to the midpoint of $CD$ . The eight resulting lines together bound an octagon inside the square (see figure). What is the area of this octagon?

Solution

$\frac{2}{3}$

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