Hilbert space satisfies the following condition: if two triangles $\triangle ABC$, $\triangle A\_1B\_1C\_1$ have equal sides lengths: $|AB|=|A\_1B\_1|$, $|BC|=|B\_1C\_1|$, $|AC|=|A\_1C\_1|$ they also have equal medians lengths: $|AM|=|A\_1M\_1|$, where $M=(B+C)/2$, $M\_1=(B\_1+C\_1)/2$. Are there other Banach spaces with this property? (If yes, there are 2-dimensional examples, and the next question is complete classification. But I doubt.)



Equality of triangles in normed spaces
Firstly, notice that you may apply your condition to the triangles $ABM$ and $A\_1B\_1M\_1$ and iterate this process. So, e.g.. for every points $X$ and $X\_1$ on $BC$ and $B\_1C\_2$ with $BX/BC=B\_1X\_1/B\_1C\_1=k/2^n$ with integer $k,n$ we obtain $AX=A\_1X\_1$. Since the binary rationals are dense on $[0,1]$, the same holds for every ratio. Similarly, we may apply this to other sides and hence see that the affine transform mapping $ABC$ to $A\_1B\_1C\_1$ is metric-preserving on the whole triangle $ABC$. THis already should suffice.


Indeed, consider an equilateral triangle $ABC$; by an affine transform we may assume that it is equilateral also in the Euclidean metric. Take a triangle $A'B'C'$ inscribed into $ABC$, with $A'$, $B'$, $C'$ dividing the sides of $ABC$ at the same ratio. Then $A'B'C'$ is also regular (in both metrics), and under the affine transform mapping $ABC$ to $A'B'C'$ the metric scales at the coefficient $\mu$ depending only on the ratio at which the vertices of $A'B'C'$ divide the sides of $ABC$.


If the sides of $A'B'C'$ are rotated at $\pi/n$ with respect to the sides of $ABC$ (angles are also Euclidean) then we may iterate inscribing such triangles $n$ times getting a triangle homothetic to $ABC$ with a known ratio. Since the metric on this triangle is $\mu^n$ times the initial one, we obtain that $\mu$ is the same for both metrics. Thus the ratios of segments rotated at $\pi/n$ is the same in our metric and Euclidean one. This is what we want.


For passing to an arbitrary space, see Anton Petrunin's answer...


