Question

ok im stumped. im looking for an equilateral triangle on the cartesian plane such that the coordinates of the vertices are all integers .

Answer 1

what are the vertices?

Answer 2

they are not given, they are for you to find

Answer 3

i guess it can be (0,0) (0,0) (0,0)
an equilateral triangle wid side length=0

Answer 4

no
thats a trivial case

Answer 5

ok, non trivial ?

Answer 6

hmm, maybe it is imposslbe.

Answer 7

hi

Answer 8

The fact that the altitude of the triangle is going to fall within the midpoint of one of the sides, and the midpoint must be half of the distance from a given set of points, I dont think it is possible to have all 6 points being integers in normal euclidian space... I don't know a good way of articulating this point... lol

Answer 9

, there are 3 points,

Answer 10

yes, but six co-ordinate points.

Answer 11

, right, the midpoint is not a problem, for instance you can have 2 as your base, and half of it is 1

Answer 12

here is what i did , ok i realised that (0,0) (x,0) and (1/2 x , sqrt3 / 2 x ) wont work

Answer 13

Yes, but then your altitude will give that other point at 1/2 sqrt 3 * a
which is irrational...

Answer 14

so lets look at rotating this equilateral triangle

Answer 15

right, i agree , so lets rotate it

Answer 16

http://mathworld.wolfram.com/EquilateralTriangle.html

the first set of equations here kind of make the point im trying to.

Answer 17

so imagine we have a circle, one vertex will be the center,
the other two will land on the circle, and make 60 degrees .

Answer 18

(0,0), ( x1,y1) (x2,y2)

Answer 19

, you know i would hate to find out this is impossible problem,

Answer 20

here Given the distances of a point from the three corners of an equilateral triangle, , , and , the length of a side is given by


(16)
(Gardner 1977, pp. 56-57 and 63). There are infinitely many solutions for which , , and are integers. In these cases, one of , , , and is divisible by 3, one by 5, one by 7, and one by 8 (Guy 1994, p. 183).

Answer 21

I'm fairly sure that because of the law of sines and the fact that they want this in normal euclidian space that it is unable to be done... but I cant think of a good proof for you.

Answer 22

looks like wolfram says there are infintie solutions

Answer 23

No, that is Napoleon's theory... it deals with creating eq triangles from the sides of any given trigangle..

Answer 24

That is a rather different question.

Answer 25

hmmm, well hero posted it

Answer 26

http://www.ams.org/samplings/feature-column/fcarc-taxi
about half way down they have a solution... but it does seem to violate a few of euclids axioms.