Loci and Concurrency Introduction

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Sometimes it is necessary in geometry to specify the location of all points satisfying one or more conditions. For example, location of set of points equidistant from two given points, location of set of points equidistant from a given point etc. This is done by considering a set of points that satisfies the given condition(s) and then testing that every point that satisfies the given condition(s) is in the set. Here are a few examples for locating set of points that satisfy the given condition(s).

Example 1:

Mark a point O on a sheet of paper. Mark another point on the same sheet of paper such that OA=2.5 cm (or any convenient distance).

Now mark B, C, D, E, F ………… all coplanar with O and A and also such that each of them is at a distance of 2.5cm from O. Join all these points. What is the pattern of the geometric figure that you get?

You get a closed curve whose points on it are equidistant from O. It is a circle. Thus “a circle is a set of points equidistant from a given point”.

A circle satisfies the condition set of all points equidistant from a given point.

We call a set of points satisfying given condition(s) as the locus of points.

Thus, we can define a circle as the locus of all points equidistant from a fixed point.

Example 2:

Consider two parallel lines l and m. Mark points A, B, C, D equidistant(d) from both l and m. Go on adding more and more such points. What is the pattern of the geometric figure got by joining all points? (A, B, C, D……)

The pattern formed by all points which are equidistant from both l and m is a line (n). The set of all points on this line n have the common property of being equidistant from both l and m. Hence, this line (n) is called the locus of the points that are equidistant form two parallel lines l and m.

Observe that the locus of points equidistant from l and m (parallel lines) is another line n which is at a constant distance d>0 from both l and m and that every point is in the same plane and a distance of d from both l and m.