See also: Mediator

In plane geometry, the mediating of a segment is the whole of the equidistant points of the ends of the segment.

This unit is the line passing by the Milieu of the segment and which is Perpendiculaire to the segment .

The mediator of a segment thus divides the plan is two half-planes. The points closer to has than B are in one of the half-planes and the points closer to B than of has are in the different one. Thus the borders of the Diagramme of Voronoï are segments of mediating.

Properties

In a triangle, the mediating ones on the three sides are convergent in a point which is the center of the circle circumscribed of this triangle.

The mediator of a segment is a axis of symmetry of this segment. In a Right-angled , the mediating ones on the sides are also axes of symmetry of the rectangle.

Construction with the compass and the rule

This construction is allotted to Œnopide de Chios. It makes it possible to build the mediator of one segment using a rule and a compass. One thus does not use a square or of scale.

That is to say the segment. One adjusts initially the compass with an unspecified ray, higher than half length AB. With this spacing of compass, one traces a circle centered on has, then a of the same circle radius centered on B. These two circles are cut in two points C and D. One plots finally the straight line (CD) which is the mediator of.

Indeed, like radii of the circles CA = CB and DA = dB. The points C and D are thus two points distinct from the mediator. The line passing by C and D is necessarily the mediator of.

Interest of this construction

This construction is usually privileged of share its best precision in comparison with the use of the rule and the square since it is not necessary to measure the segment to find the medium of it.

This construction makes it possible to trace the perpendicular on a given line passing by a given point. Thus any construction with the square is considered realizable with the rule and the compass only.

Let us consider a line (D) and a point C external on this line. One starts by tracing a circle of center C which will cut the line (D) in two points has and B. Thanks to preceding construction, one builds the mediator of. As C is at equal distance from has and B, C is on this mediator. Thus the mediator of is the line perpendicular to (D) and passing by C.

Mediator plan

In Solid geometry the mediator plan of a segment or a bipoint (has, B) is the whole of the points of space which are equidistant of has and B. By using the scalar product, one shows that this unit is the plan passing by the medium of and orthogonal with right-hand side (AB).

More generally, one can in the same way define in an Euclidean space the hyperplane mediator of a bipoint.

See too

Internal bonds

• Triangle
• List of the remarkable elements of a triangle

External bonds

• basic Vocabulary of the Geometry

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