# Homework Help: 6 sided shape drawing

1. Feb 17, 2015

### Gary Smart

1. The problem statement, all variables and given/known data
The problem involves drawing a particular set of lines and analysing anything of interest. The steps are as follows:
: Begin with a hexagon with standard notation.
: Draw a line (T) that passes through point B and which is parallel to AC.
: Produce the lines AD and EC, remember to jot X at the intersection point.
: Create line NX, with N on line (T) and NX is right angles to T.
: Produce a circle with XB as the diameter.

3. The attempt at a solution
I have attached a copy of my drawing, it doesn't have the construction lines because it is only rough, i'm firstly interested to see if I have the general idea.

Gary.

#### Attached Files:

• ###### Hexagon1.jpg
File size:
10 KB
Views:
196
2. Feb 17, 2015

### Staff: Mentor

Looks good so far.

You could label T and mark the right angle at N.

3. Feb 18, 2015

### Campbell1982

Thank you for your help mfb. I have drawn the hexagon and have attached a copy. I have labelled T and marked the right angle at N. I'm interested also in finding anything interesting about the circle. The only thing I can see really is the number of triangles produced inside of it and that it is in contact with two vertices.

#### Attached Files:

• ###### Hexagon 2.jpeg
File size:
21.1 KB
Views:
194
4. Feb 18, 2015

### Staff: Mentor

The circle seems to go through N and C, it would be interesting to see if this is a coincidence.
Does the hexagon have to be regular as in your second sketch?

5. Feb 18, 2015

### Campbell1982

The hexagon has to be regular.

6. Feb 23, 2015

### Gary Smart

Triangle BXN because is circumscribed. XB is a diameter and the midpoint of line XB is the centre of the circle. This also means that any line constructed from line XB will be a right angled triangle (circle theorem: angles in a semi-circle). Also, I think the midpoint between A and the hexagon's centre will produce a cyclic quadrilateral. I can't find anything else of particular interest.

7. Feb 23, 2015

### Staff: Mentor

I don't understand some parts of the description but the right angle is the important part (also for C).
Me neither.

8. Feb 25, 2015

### Gary Smart

Would you agree that any line produced from line XB to the the circle's diameter will produce a right-angled triangle? What do you mean about the right angle being important and C?
Do you think there is anything of interest to point out about Euler's line and the circle?

9. Feb 26, 2015

### Staff: Mentor

I don't understand which triangle you mean.
Without the right angle, the argument does not work.
What, where?

10. Feb 27, 2015

### Gary Smart

I wrote that statement slightly wrong, what I actually meant was: Any triangle that has a point on the circumference of the circle with line XB (line XB is a diameter of the circumcircle) as one side will always produce a right-angled triangle.

What do you mean by the argument?

11. Feb 27, 2015

### Staff: Mentor

Sure, but how is that relevant?