Given lines a, b, c. If a is parallel to c, b is parallel to c, then prove the parallelism of lines a and b.

We argue consistently:

1. If lines c and b are parallel to each other, then the shortest distance between them will be a perpendicular drawn from any point from one straight line to another.

2. The same applies to straight lines a and c. Moreover, if the straight line a is between the straight lines c and b, then the perpendicular from the straight line c to the straight line b will simultaneously be the perpendicular from the straight line c to a.

3. If we are straight line a beyond the area of ​​the plane between b and c, then the picture will not change, since the perpendicular between b and c can always be extended to the straight line a. Therefore line a is parallel to b.