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This essential trains for: SAT-I, AMC-8, AMC-10, AMC-12, Math Kangaroo 7-8, Math Kangaroo 9-10.

A locus is a well defined set of points.

The plural of locus is loci.

Circle

A circle is the locus of all the points in a plane that are equally far from a given point named center.

The points that are inside the contour of the circle are not usually considered to be 'the circle'. Instead, the set of points that the circle surrounds is called a disc.

In 3D, the same definition produces a sphere.

Perpendicular bisector of a segment

The following definition is valid only in 2D (a plane):

Definition: The perpendicular bisector of a segment is the locus of the points that are equally far from the ends of the segment.

Obviously, in 3D space, the perpendicular bisector of a segment is a plane.

Construction: The perpendicular bisector of a segment can be constructed using straight edge and compass. Take a distance in the compass that exceeds half the length of the given segment. Place the needle of the compass in one end of the segment and trace a circle. Keeping the same distance in the compass, place the needle at the other end and repeat. Draw a line between the two points where the circles intersect - this is the perpendicular bisector.

Properties: An arbitrary point P on the perpendicular bisector forms an isosceles triangle with the ends A and B of the given segment. (prove this)

Properties: The perpendicular bisector is the locus of the centers of all the circles in which the given segment is a chord.

Angle bisector of an angle

The following definition is valid only in 2D (plane):

Definition: The bisector of an angle is the locus of all the points that are equally far from the edges of the angle.

Construction: The bisector of an angle can be constructed using straight edge and compass. Take a distance in the compass, place the needle at the vertex of the given angle and trace a circle. Keeping the same distance in the compass place the needle where the circle intersects one of the edges of the angle and trace an arc within the interior of the angle. Repeat for the other intersection of the circle with the angle. Draw a ray between the vertex of the angle and the intersection of the last two arcs. This is the bisector of the angle.

Properties: Any point that is equally far from the edges of the angle lies on the bisector of that angle.

Properties: All the centers of the circles that are tangent to both edges of an angle lie on the bisector of that angle.