Date: Sep 28, 2013 7:09 AM
Subject: Fermat point in 2 and 3-space
Points a,b and c are variable points on median lines OA,OB,OC of an equilateral triangle ABC with origin at O.
Angle u between any two medians is found from 2 cos(u) + 1 = 0.
Show that Oa + Ob + Oc is minimum at the origin.
Similarly generalizing into 3-space,
Points a,b,c and d are variable points on altitudes OA,OB,OC and OD of a regular tetrahedron ABCD with origin at O.
Angle u between any two altitudes is found from 3 cos(u) + 1 = 0.
Show that Oa + Ob + Oc + Od is minimum at the origin.
My motivation has been to get some imagination about a hyper-solid minimizing Oa + Ob + Oc + Od + Oe at origin in 4-space with an angle given by 4 cos(u) + 1 = 0...
Please point to any references in this direction.