Today I was learning about vectors. The teacher gave me the following problem:
Consider two regular pentagons $A_1A_2A_3A_4A_5, B_1B_2B_3B_4B_5$ on the plane. The center of the first pentagon is $O_A$, the center of the other one is $O_B$. Prove that $$\sum_{i=1}^5 \overrightarrow{A_iB_i}=5\overrightarrow{O_AO_B}$$
I can add and subtract vectors, but I can’t multiply them yet. Please help solving it, and also of you can, try to generalize!
This problem is suited to affine geometry. Given an $n$-gon $\,A_1A_2\dots A_n,\,$ the centroid is defined as the weighted sum $$O_A:=\frac1n(A_1+A_2+\dots+A_n). \tag{1}$$ The same holds for any other $n$-gon $\,B_1B_2\dots B_n.\,$ Now $$\overrightarrow{O_AO_B} := O_B-O_A = \frac1n \sum_{i=1}^n \overrightarrow{A_iB_i}. \tag{2}$$