perplexus.info :: Shapes : Solid triangles (3)

Solid triangles (3) (Posted on 2010-08-18)

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

No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)

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

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info