Simple geometric proof of parallel lines cut by transversals

Question

Three parallel lines a,b and c are cut by transversal ABC. I need to prove that, if $AB = BC$, then $A'B' = B'C'$. I've made this drawing in geogebra. Any idea of what theorem is this? Could you guys give me some help? Thanks!

Answer 1

In general, this is called "intercept theorem".

To distinguish it from the "intercept theorem (for triangles)", some call it "extended intercept theorem" or "generalized intercept theorem".

There are 2 ways of proving it:-

(1) by using properties of a parallelogram; or

(2) Using "intercept theorem (for triangles)" [twice] after joining A'C.