Hello again,

I'm currently suck on another math question. Question 1 was asked here: http://twistypuzzles.com/forum/viewtopic.php?f=7&t=18998 and you were a great help then so I hope you can help again.

Everyone knows that the distance to the center of an equilateral triangle is 1/3 of its height in from the base. At least I know this holds for a plane but is it also true for equilateral triangles formed on a sphere?

Lets say I have an equilateral triangle cut by 3 great circles. Let's call one of these the equator and the base of this triangle. If the height of the triangle is A degrees above the equator then is the center of this triangle A/3 degrees above the equator? I don't think so... if A=90 isn't the center 45 degrees above the equator or A/2? Doing this in my head without a globe handy so I'm not certain. Anyways I'm looking for the general solution for any A (I think I need to say A is less then or equal to 90 degrees).

Help,
Carl

Hint: just think of the equilateral triangle that would lie on a plane connecting the 3 points, which would slice through the sphere, and project the centre of that triangle onto the surface of the sphere.

NICE!!! I always over complicate things.

Thanks,
Carl

And I tend to over-simplify things, so I'm not even sure this will work.

A couple thinks to think about:
Depending on the size of the spherical equilateral triangle, the angles change.
So, a great circle is technically a spherical equilateral triangle.
You can have an equilateral triangle that covers more than half of the sphere.

This is true, and in that case you can just project the centre of the planar triangle through the centre of the sphere to the surface on the opposite side of the sphere.

Twelvety.

Quite true of course, but that wasn't my (completely moot ) point.
I swear I read these things, but It just never clicks what somebody is actually asking until after I answer