Question 1144575
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The Figure below illustrates the situation:


    +--------------X--------------------+

    A  --->        C             <----  B


You see the starting points A and B  and the meeting point C.


They started simultaneously at 8:00 am and move towards each other.


They met at the point C at 11:00 am spending 3 hours each.



OK and very good.


Now, there is one EXCLUSIVE case, when their moving rates are the same.  In this case, they will get the ending points A and B 
simultaneously after 3 hours (i.e. at 2:00 pm), each will turn back, and, OBVIOUSLY, will meet next time 
in 3 hours after 2:00 pm (i.e. at 5:00 pm) exactly at the midpoint between A and B.



    Memorize this case and this answer:  6 hours after 11:00 am, at 5:00 pm.



Now we consider the general case, when their moving rates are different.

    It is a miracle, but we will get THE SAME ANSWER (!)



So, now I consider the general case, when they have different moving rates . . . 


Again, I consider their movements from A to B and their meeting at the point C.


Now warp the segment AB into a (or "the") circle, by uniting points A and B.


Imagine that it is your ORIGINAL configuration and that cyclists started at the same point A=B at this circle, 
moving in opposite directions.


You will get an <U>EQUIVALENT model</U>, but now the points move not along the segment AB, but along the circle circumference.


So, they started from one single common point A=B at 8:00 am, move in opposite directions along the circle 
and met at 11:00 somewhere at the point C on the circle.


Fantastic !  Let's move forward . . . (and further . . . )


They covered the entire distance AB, which we treat now as the circumference of the circle.


So, next they started at the point C on the circle and move in opposite directions.  When they will meet again ?


But of course, the situation after 11:00 am is exactly as it was at 8:00 am: they started from the common point on the circle 
(now it is point C instead of A=B) and move in opposite directions.


When they will meet each other next time ?


-- But of source, in 3 hours, i.e. at 2:00 pm, when they cover the length of the circumference again.    


Because the situation is REPEATED.


But stop for a minute and be careful (!)


In my consideration, they met each other at 2:00 pm, but it is the meeting on the circle - not in the rectilinear segment AB.


They need to cover the total circumference ONE MORE time in 3 hours - then their next meeting at 5:00 pm will be REAL MEETING

on the segment AB after reflections at the ends (!) (!) (!)


So, the answer is: they will meet each other on the segment AB next time 6 hours after 11:00 am, i.e. at 5:00 pm.
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Answered and solved.


And explained.


And completed.


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<U>Short comment at the end</U>.


<pre>
    On the circle, they meet each other every 3 hours, by covering (together) the circumference of the circle.

    But on the segment AB, they meet each other (accounting for reflections) every 6 hours.
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