center normal and center normal plane as loci

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  $P\notin AB$.  In the triangle $PAB$, let the angle bisector of $\mathrm{\angle }P$ intersect the side (http://planetmath.org/Triangle) $AB$ in the point $D$.  Then we have

$$\mathrm{\Delta }PDA\cong \mathrm{\Delta }PDB\mathit{\hspace{1em}}\text{(SAS)},$$

whence

$$\mathrm{\angle }PDA=\mathrm{\angle }PDB={\mathrm{\hspace{0.33em}90}}^{\circ },DA=DB.$$

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  $Q\ne D$.  Then we have

$$\mathrm{\Delta }QDA\cong \mathrm{\Delta }QDB\mathit{\hspace{1em}}\text{(SAS)},$$

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	CircumCircle
Related topic	AngleBisectorAsLocus