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midpoint (Definition)

If $AB$ is a segment, then its midpoint is the point $P$ of the segment whose distances from $B$ and $C$ are equal. That is, $AP = PB$.

The midpoint of segment $AB$ can be found with ruler and compass as follows: Draw two circles with radius $AB$ and centers $A,B$ respectively. Let $P,Q$ the intersection points of the circles. Then the intersection $T$ of $PQ$ wih $AB$ is the midpoint of $AB$.

\framebox{ \begin{pspicture*}(-5,-4)(5,4) \pstGeonode[PosAngle={270,270}](-2,-2)... ...=MarkHash]{A}{T} \pstSegmentMark[SegmentSymbol=MarkHash]{B}{T} \end{pspicture*}}

There are several arguments to see why $T$ is indeed the midpoint of $AB$. Because of the circles having the same radius, $AP=AQ=BP=BQ$. It follows that $\triangle PAQ\cong\triangle PBQ$ and $\triangle BAQ\cong \triangle BAQ$ and that they all are isosceles. Then $\angle APT =\angle TPB$ and so $PT$ is the angle bisector of an isosceles triangle and thus also a median. We conclude that $T$ is the midpoint.

An alternative (yet essentially equivalent) argument is that since $AP=AQ=BP=BQ$, then $ABCD$ is a parallelogram (in fact, a rhombus) and therefore the intersection $T$ 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 $AB$ such that the ratio $\frac{\overrightarrow{AP}}{\overrightarrow{PB}}=1$.

Generalization. The notion of a midpoint can be generalized. In a geometry with the congruence axioms (such as a neutral geometry), $P$ is a midpoint of points $B$ and $C$ if $P,A,B$ are collinear and line segment $AP$ is congruent to line segment $BP$.



"midpoint" is owned by Mathprof. [ full author list (5) | owner history (3) ]
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See Also: Euler line, directed segment

Keywords:  Geometry

Attachments:
coordinates of midpoint (Result) by pahio
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Cross-references: congruent, line segment, collinear, neutral geometry, congruence axioms, geometry, ratio, contains, line, directed segments, diagonals, rhombus, parallelogram, equivalent, median, angle bisector, isosceles, arguments, intersection, centers, radius, circles, compass, ruler, distances, point, segment
There are 44 references to this entry.

This is version 16 of midpoint, born on 2001-10-29, modified 2007-09-01.
Object id is 627, canonical name is Midpoint.
Accessed 5354 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M15 (Geometry :: Real and complex geometry :: Geometric constructions)
Pending Errata and Addenda
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html vs page images by drini on 2005-01-12 14:01:11

I'm sorry that this work temproarily only works in page image for now

the problem lies in latex2html not being able to correctly parse pstricks nodes (pst-node) which is a dependency on the package I use for the euclidean drawings.

I'll try to get the picture into a eps soon and include it instead of typing directly the code.
 f
G -----> H G
p \ /_ ----- ~ f(G)
 \ / f ker f
 G/ker f 
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