hi Stefy,

In post 12, I did a similar thing or a tetrahedron. It's hard to show this in 2D. I am assuming that it is possible to do this so that the diagram looks the same whichever point (A,B,C,D or E) is "at the top". OR to prove this cannot be done.

If you can prove what you have said, then it only remains to prove that, for a solution, all rays must come from a single point. If this is so, then you have proved impossibilty.

LATER EDIT:

5 rays are drawn from a point in space

I re-read the question. So all five come from one point. proving impossibility has suddenly got easier.

EVEN LATER EDIT:

Assume it is possible.

Let O be the point the rays come from and OA,OB,OC,OD,OE be the rays.

There is no loss of generality in placing A,B,C,D,E on a sphere, since the rays can always be extended until they cut the sphere without changing the angles between them.

Let A be at the "top" of the sphere. B,C,D,E must lie in a horizontal plane since these angles are equal: AOB,AOC,AOD,AOE.

I consider three cases: (i) B,C,D,E lie in a plane that cuts O (ii) B,C,D,E lie in a plane below O (iii) B,C,D,E lie in a plane above O.

(i) angle DOB = 180; angle DOC = 90 =><=

(ii) angle AOB is obtuse.

Let DB and EC cross at F. angle BFC = 90.

O is the vertex of a square based pyramid OBCDE so BOC < 90 =><=

(iii) DOA + AOB = DOB =><=

So it is impossible.

Bob

Bob