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Midpoint
If 
is a segment, then its midpoint is the point
of the segment whose distances
from 
and 
are equal. That is, 
.
The midpoint of segment 
can be found with ruler
and compass
as follows: Draw two circles
with radius
and centers 
respectively. Let 
the intersection
points of the circles. Then the intersection 
of 
wih 
is the midpoint of 
.
...
...=MarkHash]{A}{T}
\pstSegmentMark[SegmentSymbol=MarkHash]{B}{T}
\end{pspicture*}}](http://myyn.org/static/assets/mathbrowser/article/2732/images/img14.png)
There are several arguments
to see why 
is indeed the midpoint of 
. Because of the circles having the same radius, 
. It follows that
and
and that they all are isosceles. Then
and so 
is the angle bisector
of an isosceles triangle and thus also a median. We conclude that 
is the midpoint.
An alternative (yet essentially equivalent) argument is that since 
, then 
is a parallelogram
(in fact, a rhombus) and therefore
the intersection 
of its diagonals
is the midpoint of each one.
With the notation of directed segments, the midpoint is the point on the line
that contains
such that the ratio
.
Generalization. The notion of a midpoint can be generalized. In a geometry
with the congruence axioms
(such as a neutral geometry), 
is a midpoint of points 
and 
if 
are collinear
and line segment
is congruent
to line segment 
.