## Re: [EMHL] Rational distances from the vertices of aquilateral triangle

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• Dear Antreas, I wrote ... Not always. Consider the trivial case when the point P with distances a, b, c from the vertices of the equilateral triangle, lies on
Message 1 of 13 , Aug 17 8:22 AM
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Dear Antreas,
I wrote
> in the equation
> 3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2
> the four variables are equivalent
> and we can interchange them.

Not always.
Consider the trivial case when the point P
with distances a, b, c from the vertices of the
equilateral triangle, lies on one side of the
equilateral triangle that has side length d.
A general solution of this Diophantine equation
is given by the formulas
a = m^2 + 3n^2
b = Abs((m + n)(m - 3n))
c = Abs(4mn)
d = Abs((m + 3n)(m - n))
For m = 2, n = 1 we get
a = 7, b = 3, c = 8, d = 5
Which means that the side of the equilateral
triangle can be 5, 3, 8 but it can't be 7.
It is interesting the figure.
Construct the equilateral triangle A1P2P3 with side 8
On P2P3 take P1 such that P2P1 = 5 and P1P3 = 3
and on the equilateral triangle A1P2P3 construct
two equilateral triangles A3P2P1 and A2P1P3.
We have A1P1 = A2P2 = A3P3 = 7 and the points
P1, P2, P3 are the required points for the 3 equilateral
triangles. The conclusion is that there is not fourth
equilateral triangle with side 7. The number 7 in this
tetrad is always the distance of P from the opposite vertex.
Best regards
Nikos
• Dear Nikos, ... there are equilateral tringle with side 7 and points that distfrom its vertices 3, 4 and 5. See:
Message 2 of 13 , Aug 17 10:26 AM
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Dear Nikos,

El 17/08/2012 17:22, Nikolaos Dergiades escribió:
> Dear Antreas,
> I wrote
>> in the equation
>> 3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2
>> the four variables are equivalent
>> and we can interchange them.
> Not always.
> Consider the trivial case when the point P
> with distances a, b, c from the vertices of the
> equilateral triangle, lies on one side of the
> equilateral triangle that has side length d.
> A general solution of this Diophantine equation
> is given by the formulas
> a = m^2 + 3n^2
> b = Abs((m + n)(m - 3n))
> c = Abs(4mn)
> d = Abs((m + 3n)(m - n))
> For m = 2, n = 1 we get
> a = 7, b = 3, c = 8, d = 5
> Which means that the side of the equilateral
> triangle can be 5, 3, 8 but it can't be 7.
> It is interesting the figure.
> Construct the equilateral triangle A1P2P3 with side 8
> On P2P3 take P1 such that P2P1 = 5 and P1P3 = 3
> and on the equilateral triangle A1P2P3 construct
> two equilateral triangles A3P2P1 and A2P1P3.
> We have A1P1 = A2P2 = A3P3 = 7 and the points
> P1, P2, P3 are the required points for the 3 equilateral
> triangles. The conclusion is that there is not fourth
> equilateral triangle with side 7. The number 7 in this
> tetrad is always the distance of P from the opposite vertex.
> Best regards
> Nikos
>
>

there are equilateral tringle with side 7 and points that distfrom its
vertices 3, 4 and 5. See:

http://www.xente.mundo-r.com/ilarrosa/GeoGebra/DistEnterasTriangEquil.html

From a equilateral triangle with side 8, you can construc 12 of thats
triangles of side 7.

--
Best regards,

Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosa@...
http://www.xente.mundo-r.com/ilarrosa/GeoGebra/
• Dear Ignacio, thank you very much. You are right. What I said is nonsense. I was deceived thinking that the fourth case of equilateral triangle with side 7
Message 3 of 13 , Aug 17 12:51 PM
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Dear Ignacio,
thank you very much. You are right.
What I said is nonsense.
I was deceived thinking that the fourth
case of equilateral triangle with side 7
would be a trivial case as the other three.
Best regards

> there are equilateral tringle with side 7 and points that
> distfrom its
> vertices 3, 4 and 5. See:
>
> http://www.xente.mundo-r.com/ilarrosa/GeoGebra/DistEnterasTriangEquil.html
>
> From a equilateral triangle with side 8, you can construc
> 12 of thats
> triangles of side 7.

> Best regards,
>
> Ignacio Larrosa Cañestro
> A Coruña (España)
> ilarrosa@...
> http://www.xente.mundo-r.com/ilarrosa/GeoGebra/
>
>
>
> ------------------------------------
>