Is there a solid where all triangles on the surface are isosceles? Are there any solids in $R^{3}$ for which, for any 3 points chosen on the surface, at least two of the lengths of the shortest curves which can be drawn on the surface to connect pairs of them are equal? 

If one has a plane 3-connected graph in the plane all of whose faces are triangles it is known that one can not always realize this graph by a convex 3-dimensional polyhedron all of whose faces are congruent strictly isosceles triangles. There are exactly 8 types of such graphs which can be realized with equilateral triangles - the so called convex deltahedra. It is however an open problem whether one can always realize such a graph so that all of the faces are isosceles but have faces with edges of different lengths. Your question goes beyond these considerations in its requirements.
