Re: [EMHL] Rational distances from the vertices of aquilateral triangle
- Dear Nikos,
El 17/08/2012 17:22, Nikolaos Dergiades escribió:
> Dear Antreas,there are equilateral tringle with side 7 and points that distfrom its
> 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
vertices 3, 4 and 5. See:
From a equilateral triangle with side 8, you can construc 12 of thats
triangles of side 7.
Ignacio Larrosa Cañestro
A Coruña (España)
- 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.
> there are equilateral tringle with side 7 and points that
> distfrom its
> vertices 3, 4 and 5. See:
> 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)
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