Problems

Let \(ABCDEF\) be a regular hexagon. Points \(G\) and \(H\) lie on \(EF\) and \(DE\) respectively such that \(|EG|=|EH|\). Furthermore, the area of quadrilateral \(ABGF\) is equal to the area of quadrilateral \(BGEH\), which are both equal to the area of \(BCDH\). What’s the ratio \(\frac{|EG|}{|EF|}\)?

Below is a regular octagon. Given that its side length is \(1\), what’s the difference between the area of the red rectangle and the rest of the octagon?

In the diagram, all the small squares are of the same size. What fraction of the large square is shaded?

We have a square of side length 1. At each vertex of the square, we draw a circle of radius 1. What is the area bounded by all four circles?

Let \(\triangle ABC\) be a triangle and \(D\) be a point on the edge \(BC\) so that the segment \(AD\) bisects the angle \(\angle BAC\). Show that \(\frac{|AB|}{|BD|}=\frac{|AC|}{|CD|}\).