We have #A_(((5,0))),B_(((-2,4)))#.How to prove that if #Din(AB)#,then #D# can not have both coordinates integer numbers?

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Answer 1 · 2017-06-09T21:27:28

See below.

If #D in [AB]# then #D = A+lambda(B-A)#

Now supposing that #D# has integer coordinates

#D = (m,n) = (5,0)+lambda(-7,4)# or

#lambda = (5-m)/7=n/4# or

#20-4m=7n# or

#4m+7n=20#

Now by Bezout's Lemma,

if the gcd(4,7,20) = 1, does not exist integers #m, n# such that #4m+7n=20#