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Unit lengths and triangles (Posted on 2009-09-09) Difficulty: 4 of 5
Given A=(1,0), B=(0,0), and C=(0,1)
find all unit length equilateral triangles XYZ such that the lengths AX, BY, and CZ are all one unit.

No Solution Yet Submitted by Jer    
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mathematica solution | Comment 1 of 2

let X,Y,Z be (x1,x2) (y1,y2) (z1,z2) respectively
then this boils down to solving these 6 equations
(x1-1)^2+x2^2=1
y1^2+y2^2=1
z1^2+(z2-1)^2=1
(x1-y1)^2+(x2-y2)^2=1
(y1-z1)^2+(y2-z2)^2=1
(x1-z1)^2+(x2-z2)^2=1

Tried for an analytical solution, still working it out, but for now I have Mathematica's solutions which it found these 12 (in order x,y,z)
1 (1/2,sqrt(3)/2) (1,0) (0,0)
2 (1/2,sqrt(3)/2) (-1/2,sqrt(3)/2) (0,0)
3 (1/2,-sqrt(3)/2) (1,0) (0,0)
4 (1/2,-sqrt(3)/2) (-1/2,sqrt(3)/2) (0,0)
5 (0,0) (0,1) (sqrt(3),1/2)
6 (0,0) (0,1) (-sqrt(3),1/2)
7 (0,0) (sqrt(3)/2,-1/2) (sqrt(3)/2,1/2)
8 (0,0) (-sqrt(3)/2,-1/2) (-sqrt(3)/2,1/2)
9 (1,1) (1,0) (1/2,(2-sqrt(3))/2)
10 (1,1) (1,0) (1/2,(2+sqrt(3))/2)
11 ((2-sqrt(3))/2,1/2) (1,0) (1,1)
12 ((2+sqrt(3))/2,1/2) (1,0) (1,1)


  Posted by Daniel on 2009-09-09 14:58:36
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