# Can Parallelograms Be Constructed in a Convex Hexagon?

• MHB
• maxkor
In summary, a parallelogram is a four-sided shape with two pairs of parallel sides. A hexagon is a six-sided shape with six angles. There are three parallelograms in a hexagon, formed by connecting opposite vertices. The properties of parallelograms in a hexagon include two pairs of parallel sides, equal opposite sides and angles, and supplementary consecutive angles. The sides and angles of a parallelogram in a hexagon are congruent and supplementary, respectively.
maxkor
In a convex hexagon $ABCDEF$ exist a point $M$ such that $ABCM$ and $DEFM$ are parallelograms . Prove that exists a point $N$ such that $BCDN$ and $EFAN$ are also parallelograms.

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Proof using vectors:

Let $\vec{a},\vec{b},\vec{c},\vec{d},\vec{e},\vec{f},\vec{m}$ be vectors representing the points $A,B.C,D,E,F,M$. Then $\vec{m} = \vec{c} + (\vec{a} - \vec{b})$. Therefore $$\vec{a} + \vec{c} - \vec{b} = \vec{d} + \vec{f} - \vec{e}$$ and so $$\vec{b} + \vec{d} - \vec{c} = \vec{a} + \vec{e} - \vec{f}.$$ Let $N$ be the point given by the vector $$\vec{n} = \vec{b} + \vec{d} - \vec{c}.$$ Then $N$ has the property that $BCDN$ and $EFAN$ are parallelograms.

## 1. What is a parallelogram in a hexagon?

A parallelogram in a hexagon is a four-sided shape that has two pairs of parallel sides and is contained within a larger six-sided shape called a hexagon.

## 2. How can you identify a parallelogram in a hexagon?

A parallelogram in a hexagon can be identified by looking for two pairs of parallel sides within the hexagon. These sides will be opposite each other and will have the same length.

## 3. What are the properties of a parallelogram in a hexagon?

The properties of a parallelogram in a hexagon include having two pairs of parallel sides, opposite angles that are congruent, and opposite sides that are equal in length.

## 4. How is the area of a parallelogram in a hexagon calculated?

The area of a parallelogram in a hexagon can be calculated by multiplying the base (one of the parallel sides) by the height (the distance between the base and the opposite parallel side).

## 5. What are some real-life examples of parallelograms in a hexagon?

Some real-life examples of parallelograms in a hexagon include the shape of a stop sign, the logo for the Boy Scouts of America, and the structure of a honeycomb.

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