Triangle inscribed in a Hexagon

Oct 2012
92
0
san francisco
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.
 

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Prove It

MHF Helper
Aug 2008
12,897
5,001
I don't see a picture...
 
Oct 2012
92
0
san francisco
sorry i just added it right now
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
Is it a regular hexagon or an arbitrary hexagon?
 
Oct 2012
92
0
san francisco
it is a regular hexagon
 
Nov 2007
985
175
Trumbull Ct
ABCDEF is aregular hexagon. Name the three midpoints that form the triangle
 

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)
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.
 
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Nov 2007
985
175
Trumbull Ct
There are two additional triangles which can be drawn meeting the general requirements
 
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Oct 2012
92
0
san francisco
Great thanks for your help everyone.