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# 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 $P\not\in AB$. 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”.

## Mathematics Subject Classification

51M15*no label found*51N05

*no label found*51N20

*no label found*

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