|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
center normal and center normal plane as loci
|
(Theorem)
|
|
|
Theorem 1. In the Euclidean plane, the center normal of a line segment is the locus of the points which are equidistant from the both end points of the segment.
Proof. Let $A$ and $B$ be arbitrary given distinct points.
$1^\circ.$ Let $P$ be a point equidistant from $A$ and $B$ . If $P \in AB$ , then $P$ is trivially on the center normal of $AB$ . Thus suppose that $P \not\in AB$ . In the triangle $PAB$ , let the angle bisector of $\angle P$ intersect the side $AB$ in the point $D$ . Then we have $$\Delta PDA \;\cong\; \Delta PDB \quad \mbox{(SAS)},$$ whence $$\angle PDA \;=\; \angle PDB \;=\; 90^\circ, \quad DA \;=\; DB.$$ Consequently, the point $P$ is always on the center normal of $AB$ .
$2^\circ.$ Let $Q$ be any point on the center normal and $D$ the midpoint of the line segment $AB$ . We can assume that $Q \neq D$ . Then we have $$\Delta QDA \;\cong\; \Delta QDB \quad \mbox{(SAS)},$$ implying that $$QA \;=\; QB.$$ Thus $Q$ is equidistant from $A$ and $B$ .
Theorem 2. In the Euclidean space, the center normal plane of a line segment is the locus of the points which are equidistant from the both end points of the segment.
Proof. Change ``center normal'' in the preceding proof to ``center normal plane''.
|
"center normal and center normal plane as loci" is owned by pahio.
|
|
(view preamble | get metadata)
Cross-references: center normal plane, Euclidean space, midpoint, intersect, angle bisector, triangle, proof, segment, end points, points, locus, line segment, center normal, Euclidean plane, theorem
There is 1 reference to this entry.
This is version 9 of center normal and center normal plane as loci, born on 2009-02-10, modified 2009-02-11.
Object id is 11616, canonical name is CenterNormalAndCenterNormalPlaneAsLoci.
Accessed 910 times total.
Classification:
| AMS MSC: | 51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry) | | | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) | | | 51M15 (Geometry :: Real and complex geometry :: Geometric constructions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
