MHB Coordinates of Hexagon Vertices in Base (AC, AD)

AI Thread Summary
The discussion focuses on determining the coordinates of the vertices AB, AE, and AF of a regular hexagon in the base defined by points AC and AD. The user initially struggles with the relationships between the vertices and their coordinates but eventually clarifies that the coordinates of points A, B, C, D, E, and F are based on trigonometric calculations using a unit circle. The correct coordinates are derived as A=(1,0), B=(1/√2, 1/√2), C=(-1/√2, 1/√2), D=(-1,0), E=(-1/√2, -1/√2), and F=(1/√2, -1/√2). The user successfully calculates the coordinates for AB, AE, and AF but encounters difficulties in expressing them relative to the base coordinates. The discussion concludes with the user seeking assistance in finalizing the coordinates based on the established base.
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Consider a regular hexagon ABCDEF (in order counterclockwise). Determine the coordinates of AB, AE AND AF (->) in the base (AC, AD) (->)

AB(->)=(_____,_____)
AE(->)=(_____,_____)
AF(->)=(_____,_____)

what I mean with exemple AF(->) positive way from A to F. I have draw a it but I got problem to rewrite
I got AE=EF+FA(->) I am correct?
 
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Re: hexagon, coordinates, base

3_3_2_geo.jpg
I could uppload picture from internet but it should be insted of F it should B. In ourder counterclockwise. Well I can rewrite AE=EF+FA (->)
 
Re: hexagon, coordinates, base

No, you have the order wrong: AE= AF+ FE.
 
Re: hexagon, coordinates, base

HallsofIvy said:
No, you have the order wrong: AE= AF+ FE.
Yeah I forgot to Edit. How do I do for AF and AB? Then I got Also My base.
 
Re: hexagon, coordinates, base

View attachment 725
So I draw it.
We know from origo to A it is 1 and origo to D it is 1.
 

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I think I have thought wrong...
We got the base AC and AD and I put $$AC=(1,0)$$ and $$AD(0,1)$$
I start with AB
We know that $$AC=AB+BC$$ ( ->) that means $$AB=AC-BC <=> AB=AC-BC$$ and we know that $$AC=(1,0)$$ That means $$AB=(1,0)-BC$$ But is BC same as from A to origo?
 
So I did wrong... After a lot reading I think I got correct progress now...
We know it says regular hexagon that means from origo to any point got the length 1.
A circle is 360 degree and we got 8 lines. $$\frac{360}{8}=45$$ So we got now (look picture). we know x value is cos and y value is sin so we know
$$A=(1,0)$$
$$B=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$$
$$C=(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$$
$$D=(-1,0)$$
$$E=(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$$
$$F=(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$$
That means
$$AC=(-\frac{1}{\sqrt{2}}-1,\frac{1}{\sqrt{2}})$$
$$AD=(-2,0)$$
----
$$AB=(\frac{1}{\sqrt{2}}-1,\frac{1}{\sqrt{2}})$$
$$AE=(-\frac{1}{\sqrt{2}}-1,-\frac{1}{\sqrt{2}})$$
$$AF=(\frac{1}{\sqrt{2}}-1,-\frac{1}{\sqrt{2}})$$
So now I got all point but got problem to determine our cordinate with our base. View attachment 727
 

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