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6 Jul 2007 21:40:27 IST
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Prove that a triangle having integral coordinates can never be an equilateral triangle..
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-RAhul |
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6 Jul 2007 21:45:29 IST
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Plz answer Ill rate u.
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-RAhul |
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6 Jul 2007 21:47:34 IST
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let the coordinates of the triangle be (a,b) , (c,d) , (e,f)
let us assume the triangle is equilateral with a side length 'l' .
now the area of triangle(A) = 1/2 | a(d-f) + c(f-b) + e(b-d) |
if the coordinates area intergral then A is integral(rational).
whereas the area of an equilateral triangle is also given by(A') = sqrt(3)/4 (l)^2
now A = A'
here, LHS is rational but RHS is irrational (bcoz' of root(3) ) wich is not possible.
so, the equilateral triangle cannot have integral coordinates..
i hope this helps...
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