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Solid triangles (3) (Posted on 2010-08-18) Difficulty: 3 of 5
Three points are located on the surface of the ellipsoid: 2x2 + 2y2 + z2 = 3. One has a x coordinate of 0, another has a y coordinate of 0, and the last has a z coordinate of 0.

What is the largest possible equilateral triangle (in terms of area) that can be made using these three points as the corners? How many distinct equilateral triangles of this size are possible?

No Solution Yet Submitted by K Sengupta    
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Some Thoughts re: Parametric Thoughts Comment 4 of 4 |
(In reply to Parametric Thoughts by Brian Smith)

Rearranging the expressions for D^2 yields:

sin t = (4*(cos u)*(sin v) - (sin v)^2)/(2*(cos v))
cos t = (4*(cos u)*(sin v) - (cos u)^2)/(2*(sin u))

I tried putting these into the identity (sin t)^2 + (cos t)^2 = 1 but could not get the equation to simplify.  

So I tried some brute force number crunching and found this triangle: (0.86983, 0.86220, 0), (-1.08331, 0, 0.80800), (0, -1.04211, -0.90996).  These correspond to t=0.78099, u=5.19772, v=3.69476.  The triangle has sides 2.282763, 2.282768, 2,282748 - very close to an actual equilateral triangle.

Edited on February 21, 2016, 5:54 pm
  Posted by Brian Smith on 2016-02-21 13:38:41

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