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How can I prove this :

Let a regular hexagon ABCDEF and M, N, R and S the respective midpoints of AB, CD, DE and FA: i) Prove that the MNRS is a rectangle. ii) compare the area of MNRS and the area of ​​the hexagon

My ideas

For the first question, I proved that the diagonals MR and NS are equal and thus form a rectangle.

For the second question, I have no idea... Do you have any clues on how to prove?

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up vote 1 down vote accepted

Hint: Considering the triangle and trapezoid that are created between the hexagon and rectangle, use trigonometry and special triangles to get the width and height of the rectangle in terms of the side length. Then get the area of the hexagon in terms of the side length (split it into 6 triangles), and compare.

Other hint: Splitting the hexagon into 6 triangles, note that half of the diagonal is the height of one such triangle. Use this knowledge and the angles the diagonals make at the centre of the rectangle (which you can find with the aforementioned triangles) to get the length and width of the rectangle in terms of the side length, then proceed as above.

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