Points a, b, and c are collinear. points m and n are the midpoints of segments ab and ac. prove that bc = 2mn

Hello ! (I attached a picture of the drawing)

First, we draw a line with 3 points B,A,C in this order.Then we will note the middle of AB with M and also the middle of AC with N.

We can notice that BC = AB + AC

M - the middle of AB ⇒ AM=MB=AB/2
N - the middle of AC ⇒ AN=NC=AC/2


We can write MN as AM+AN , which means AB/2 + AC/2 . We can also write AB as BC-AC , so we will have :

MN = AB/2 + AC/2
MN = (BC-AC)/2 + AC/2
MN = BC/2 - AC/2 + AC/2 ( AC/2 with -AC/2 are reduced)
MN = BC/2    or    BC = 2·MN