Loci

The principle of equidistance is the logical foundation for most loci. It implies that for any point $P$ on the locus, the distance $d(P, Reference_1) = d(P, Reference_2)$.

Loci often involve compound conditions, where a point must satisfy multiple rules simultaneously. The resulting locus is the intersection of the individual loci defined by each rule.

In coordinate geometry, a locus is expressed as an algebraic equation. For example, the locus of points $(x, y)$ at distance $r$ from the origin is $x^2 + y^2 = r^2$.

When a condition uses inequalities (e.g., 'distance less than $5$'), the locus describes a shaded region rather than a single line or curve.

Geometric Construction: This method uses a compass and a straightedge to draw precise loci. Arcs are used to find points of equidistance, such as the intersection of two arcs to find a perpendicular bisector.

Algebraic Derivation: By assigning coordinates to fixed points and using the distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, one can derive the equation of the locus through algebraic simplification.

Step-by-Step Construction: To find the locus equidistant from two lines, first draw arcs of equal radius from the intersection point along both lines, then use those points to draw two more intersecting arcs to find the bisector path.

Region Shading: When dealing with 'at most' or 'less than' constraints, identify the boundary locus first, then test a point to determine which side of the boundary satisfies the inequality.

Boundary vs. Region: A condition stating 'distance is equal to $d$' results in a line (boundary), while 'distance is less than $d$' results in an area (region).

Inclusive vs. Exclusive: In regions, 'less than' ($<$) is represented by a dashed boundary line, while 'less than or equal to' ($\le$) is represented by a solid boundary line.

Check the Endpoints: When drawing a locus at a fixed distance from a line segment, students often forget to draw the semi-circular caps at the ends of the segment.

Multiple Constraints: If a question provides two conditions, draw both loci separately first. The final answer is usually the point(s) where they intersect or the overlapping shaded area.

Accuracy in Construction: Always keep the compass width constant when drawing arcs for bisectors. Changing the width mid-process will result in an incorrect path.

Sanity Check: After shading a region, pick a random point within that region and manually check if it satisfies all the original rules given in the problem.