If two lines have the same midpoint then the lines...

bisect each other?

that's not true if they are both segments of the same line though.

Lines have infinite length

We should be talking about line segments 

Then the segments have equal length

To see this.......draw a circle of any radius  with center O and any two diameters in the circle

Call the diameters AB   and CD...and these are equal

But  AO = BO = CO = DO  = radii

So....the radii are equal....and since O is the center.......the diameters have the same midpoint