center normal and center normal plane as loci


Theorem 1.  In the Euclidean planeMathworldPlanetmath, the center normal of a line segmentMathworldPlanetmath is the locus of the points which are equidistant from the both end pointsPlanetmathPlanetmath of the segment.

Proof.  Let A and B be arbitrary given distinct points.

1.  Let P be a point equidistant from A and B.  If  PAB,  then P is trivially on the center normal of AB.  Thus suppose that  PAB.  In the triangleMathworldPlanetmath PAB, let the angle bisectorMathworldPlanetmath of P intersect the side (http://planetmath.org/Triangle) AB in the point D.  Then we have

ΔPDAΔPDB(SAS),

whence

PDA=PDB= 90,DA=DB.

Consequently, the point P is always on the center normal of AB.

2.  Let Q be any point on the center normal and D the midpointMathworldPlanetmathPlanetmathPlanetmath of the line segment AB.  We can assume that  QD.  Then we have

ΔQDAΔQDB(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”.

Title center normal and center normal plane as loci
Canonical name CenterNormalAndCenterNormalPlaneAsLoci
Date of creation 2013-03-22 18:48:51
Last modified on 2013-03-22 18:48:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 51M15
Classification msc 51N05
Classification msc 51N20
Synonym center normal as locus
Synonym center normal plane as locus
Related topic SAS
Related topic CircumCircleMathworldPlanetmath
Related topic AngleBisectorAsLocus