The extra segment in the 8086/8088 is a 64kb region of memory that is indexed by the displacement address of the destination of certain string operations, relative to DI. Contrast that with the data segment, which is a 64kb region of memory that is indexed by the displacment address of the displacement of most operand addresses.
The stack segment is similar, but it is used for stack oriented data, relative to SP or BP.
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ray and segment
a line with two arrows on the end.
There are only three endpoint given and these are not sufficient to define a segment of a line.
it intersects the segment joining the centers of two circles
If you mean a line segment, then yes, every line segment has a midpoint. However, some ideas of a line define it as going on forever in a certain plane, so if is it a line without beginning or end, then it can't really have a midpoint since there is nothing to measure from. You would have to define a starting and ending point before it could have a midpoint.