Recognizing the shape of a triangle from afar

We discuss a question when a simple planar shape is determined by a few of its geometric or physical characteristics. The positive answer will be given for remote temperatures of sides of a triangle and remote temperatures of sides of a parallelogram. These remote temperatures are the harmonic measures of sides of a polygon evaluated at ∞. In particular, we prove that for any positive numbers ω1, ω2 and ω3 such that 0 < ω1 ≤ ω2 ≤ ω3 < ω1 +ω2, ω1+ω2+ω3=1 there is a triangle T, unique up to linear transformations, with the remote temperatures of its sides a, b and c equal to ω1, ω2 and ω3, respectively. Our results suggest several interesting relationships between the triangle's parameters, some of which may not have been discovered previously. A few open problems also will be discussed.