Yes, that's the last case: any point p on the base of an isosceles triangle with apex < pi/3 equally minimizes the sum of distances from p to the sides of the triangle. I like Fred's coinage of inceeding. I would also like to coin words for each connected component of non-equilateral isosceles triangles: Let the elongated equilaterals be called "sharp" and the squashed ones be called "flat". --Dan Fred wrote: << For isosceles triangles T with apical angle inceeding pi/3 --- exasperated at having no contrapositive for "exceeding", I hereby coin one --- any point along the base serves as centre p. For collinear T (with area zero) any point on the line serves as p.

...

"Things are seldom what they seem." --W.S. Gilbert