MathDB

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.
(a) A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio.
(b) Suppose, a square

ABCD

is inscribed in the hexagon such that

A

and

C

are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon.
(c) Suppose, the side of the hexagon is of length

1

. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.
(d) Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

6

Problems(1)