# Triangle inscribed in a Hexagon

#### gfbrd

Hey sorry if this is in the wrong place and if the picture is too small but I need some help

What can you say about a triangle that is inscribed in a hexagon, where the vertices of the triangle hits the midpoint of 3 sides of a hexagon?
if there is something about it please explain it to me so I can understand better, thanks.

#### Attachments

• 78 KB Views: 9

#### Prove It

MHF Helper
I don't see a picture...

#### gfbrd

sorry i just added it right now

#### Prove It

MHF Helper
Is it a regular hexagon or an arbitrary hexagon?

#### gfbrd

it is a regular hexagon

#### bjhopper

ABCDEF is aregular hexagon. Name the three midpoints that form the triangle

#### Soroban

MHF Hall of Honor
Hello, gfbrd!

What can you say about a triangle that is inscribed in a regular hexagon,
where the vertices of the triangle are the midpoint of 3 sides of a hexagon?

What do they want me to say?
. . It has three sides.
. . It has three angles.
. . It is equilateral.
. . It has 60o angles.

Okay, I'll get serious . . .

A regular hexagon is composed of six equilateral triangles of side a.
Consider the upper half of the hexagon.
Code:
            : - - a - - :
*  *  *  *  *
* .         . *
a *   .       .   * a
*=================*
*       .   .       *
*         . .         *
*  *  *  *  *  *  *  *  *
: - - a - - : - - a - - :
We have an isosceles trapezoid.
The side of the triangle is the median.
Its length is the average of the lengths
. . of the two parallel sides.
Hence, the side of the triangle is $$\displaystyle \tfrac{3}{2}a.$$

The triangle's perimeter is $$\displaystyle \tfrac{3}{4}$$ of the hexagon's perimeter.
The triangle's area is $$\displaystyle \tfrac{3}{8}$$ of the hexagon's area.

1 person

#### bjhopper

There are two additional triangles which can be drawn meeting the general requirements

1 person

#### gfbrd

Great thanks for your help everyone.