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Topic: Fermat point in 2 and 3-space
Replies: 0

 Narasimham Posts: 356 Registered: 9/16/06
Fermat point in 2 and 3-space
Posted: Sep 28, 2013 7:09 AM

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.

Regards
Narasimham

PostScript:

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.