Locus
11.1 Determining a Locus
Locus, in Latin, means location. The plural is loci. A locus of points is the set of points, and only those points, that satisfy given conditions.
Thus, the locus of points that are 1 in from a given point P is the set of points 1 in from P. These points lie on a circle with its center at P and a radius of 1 in, and hence this circle is the required locus (Fig. 11-1). Note that we show loci as long-short dashed figures.
Fig. 11-1
To determine a locus, (1) state what is given and the condition to be satisfied; (2) find several points satisfying the condition which indicate the shape of the locus; then (3) connect the points and describe the locus fully.
All geometric constructions require the use of straightedges and compasses. Hence if a locus is to be constructed, such drawing instruments can be used.
11.1A Fundamental Locus Theorems
PRINCIPLE 1: The locus of points equidistant from two given points is the perpendicular bisector of the line segment joining the two points (Fig. 11-2).
Fig. 11-2
PRINCIPLE 2: The locus of points equidistant from two given parallel lines is a line parallel to the two lines and midway between them (Fig. 11-3).
Fig. 11-3
PRINCIPLE 3: The locus of points equidistant from the sides of a given angle is the bisector of the angle (Fig. 11-4).
Fig. 11-4
PRINCIPLE 4: The locus of points equidistant from two given intersecting lines is the bisectors of the angles formed by the lines (Fig. 11-5).
Fig. 11-5
PRINCIPLE 5: The locus of points equidistant from two concentric circles is the circle concentric with the given circles and midway between them (Fig. 11-6).
Fig. 11-6
PRINCIPLE 6: The locus of points at a given distance from a given point is a circle whose center is the given point and whose radius is the given distance (Fig. 11-7).
Fig. 11-7
PRINCIPLE 7: The locus of points at a given distance from a given line is a pair of lines, parallel to the given line and at the given distance from the given line (Fig. 11-8).
Fig. 11-8
PRINCIPLE 8: The locus of points at a given distance from a given circle whose radius is greater than that distance is a pair of concentric circles, one on either side of the given circle and at the given distance from it (Fig. 11-9).
Fig. 11-9
PRINCIPLE 9: The locus of points at a given distance from a given circle whose radius is less than the distance is a circle, outside the given circle and concentric with it (Fig. 11-10). (If r 5 d, the locus also includes the center of the given circle.)
Fig. 11-10
SOLVED PROBLEMS
11.1 Determining loci
Determine the locus of (a) a runner moving equidistant from the sides of a straight track; (b) a plane flying equidistant from two separated aircraft batteries; (c) a satellite 100 mi above the earth; (d) the furthermost point reached by a gun with a range of 10 mi.
Solutions
See Fig. 11-11.
(a) The locus is a line parallel to the two given lines and midway between them.
(b) The locus is the perpendicular bisector of the line joining the two points.
(c) The locus is a circle concentric with the earth and of radius 100 mi greater than that of the earth.
(d) The locus is a circle of radius 10 mi with its center at the gun.
Fig. 11-11
11.2 Determining the locus of the center of a circle
Determine the locus of the center of a circular disk (a) moving so that it touches each of two parallel lines; (b) moving tangentially to two concentric circles; (c) moving so that its rim passes through a fixed point; (d) rolling along a large fixed circular hoop.
Solutions
See Fig. 11-12.
Fig. 11-12
(a) The locus is a line parallel to the two given lines and midway between them.
(b) The locus is a circle concentric with the given circles and midway between them.
(c) The locus is a circle whose center is the given point and whose radius is the radius of the circular disk.
(d) The locus is a circle outside the given circle and concentric to it.
11.3 Constructing loci
Construct (a) the locus of points equidistant from two given points; (b) the locus of points equidistant from two given parallel lines; (c) the locus of points at a given distance from a given circle whose radius is less than that distance.
Solutions
See Fig. 11-13.
Fig. 11-13
11.2 Locating Points by Means of Intersecting Loci
A point or points which satisfy two conditions may be found by drawing the locus for each condition. The required points are the points of intersection of the two loci.
SOLVED PROBLEMS
11.4 Locating points that satisfy two conditions
On a map locate buried treasure that is 3 ft from a tree (T) and equidistant from two points (A and B) in Fig. 11-14.
Fig. 11-14
The required loci are (1) the perpendicular bisector of 
and (2) a circle with its center at T and radius 3 ft. As shown, these meet in P1 and P2, which are the locations of the treasure.
Note: The diagram shows the two loci intersecting at P1 and P2. However, there are three possible kinds of solutions, depending on the location of T with respect to A and B:
1. The solution has two points if the loci intersect.
2. The solution has one point if the perpendicular bisector is tangent to the circle.
3. The solution has no points if the perpendicular bisector does not meet the circle.
11.3 Proving a Locus
To prove that a locus satisfies a given condition, it is necessary to prove the locus theorem and its converse or its inverse. Thus to prove that a circle A of radius 2 in is the locus of points 2 in from A, it is necessary to prove either that
1. Any point on circle A is 2 in from A.
2. Any point 2 in from A is on circle A (converse of statement 1).
or that
1. Any point on circle A is 2 in from A.
2. Any point not on circle A is not 2 in from A (inverse of statement 1).
These statements are easily proved using the principle that a point is outside, on, or inside a circle according as its distance from the center is greater than, equal to, or less than the radius of the circle.
SOLVED PROBLEM
11.5 Proving a locus theorem
Prove that the locus of points equidistant from two given points is the perpendicular bisector of the segment joining the two points.
Solution
First prove that any point on the locus satisfies the condition:
PROOF:
Then prove that any point satisfying the condition is on the locus:
PROOF:
SUPPLEMENTARY PROBLEMS
11.1. Determine the locus of
(11.1)
(a) The midpoints of the radii of a given circle
(b) The midpoints of chords of a given circle parallel to a given line
(c) The midpoints of chords of fixed length in a given circle
(d) The vertex of the right angle of a triangle having a given hypotenuse
(e) The vertex of an isosceles triangle having a given base
(f) The center of a circle which passes through two given points
(g) The center of a circle tangent to a given line at a given point on that line
(h) The center of a circle tangent to the sides of a given angle
11.2. Determine the locus of
(11.1)
(a) A boat moving so that it is equidistant from the parallel banks of a stream
(b) A swimmer maintaining the same distance from two floats
(c) A police helicopter in pursuit of a car which has just passed the junction of two straight roads and which may be on either one of them
(d) A treasure buried at the same distance from two intersecting straight roads
11.3. Determine the locus of (a) a planet moving at a fixed distance from its sun; (b) a boat moving at a fixed distance from the coast of a circular island; (c) plants laid at a distance of 20 ft from a straight row of other plants; (d) the outer extremity of a clock hand.
(11.1)
11.4. Excluding points lying outside rectangle ABCD in Fig. 11-15, find the locus of points which are
(11.1)
(a) Equidistant from 
and 
(b) Equidistant from and 
(c) Equidistant from A and B
(d) Equidistant from B and C
(e) 5 units from
(f) 10 units from
(g) 20 units from
(h) 10 units from B
Fig. 11-15
11.5. Find the locus of points in rhombus ABCD in Fig. 11-16, which are equidistant from (a) and ; (b) and ; (c) A and C; (d) B and D; (e) each of the four sides.
(11.1)
Fig. 11-16
11.6. In Fig. 11-17, find the locus of points which are on or inside circle C and
(11.1 and 11.2)
(a) 5 units from O
(b) 15 units from O
(c) Equidistant from circles A and C
(d) 10 units from circle C
(e) 10 units from circle A
(f) 5 units from circle B
(g) The center of a circle tangent to circles A and C
Fig. 11-17
11.7. Determine the locus of the center of (a) a coin rolling around and touching a smaller coin; (b) a coin rolling around and touching a larger coin; (c) a wheel moving between two parallel bars and touching both of them; (d) a wheel moving along a straight metal bar and touching it.
(11.2)
11.8. Find the locus of points that are in rectangle ABCD of Fig. 11-18 and the center of a circle
(11.2)
(a) Tangent to and
(b) Tangent to and
(c) Tangent to and 
(d) Of radius 10, tangent to
(e) Of radius 20, tangent to
(f) Tangent to at G
Fig. 11-18
11.9. Locate each of the following:
(11.4)
(a) Treasure that is buried 5 ft from a straight fence and equidistant from two given points where the fence meets the ground
(b) Points that are 3 ft from a circle whose radius is 2 ft and are equidistant from two lines which are parallel to each other and tangent to the circle
(c) A point equidistant from the three vertices of a given triangle
(d) A point equidistant from two given points and equidistant from two given parallels
(e) Points equidistant from two given intersecting lines and 5 ft from their intersection
(f) A point that is equidistant from the sides of an angle and 
in from their intersection
11.10. Locate the point or points which satisfy the following conditions with respect to ΔABC in Fig. 11-19:
(11.4)
(a) Equidistant from its sides
(b) Equidistant from its vertices
(c) Equidistant from A and B and from and
(d) Equidistant from and 
and 5 units from C
(e) 5 units from B and 10 units from A
Fig. 11-19
11.11. Excluding points lying outside square ABCD in Fig. 11-20, how many points are there that are
(11.4)
(a) Equidistant from its vertices
(b) Equidistant from its sides
(c) 5 units from E and on one of the diagonals
(d) 5 units from E and equidistant from and
(e) 5 units from 
and equidistant from and
(f) 20 units from A and 10 units from B
Fig. 11-20
11.12. Prove that the locus of points equidistant from the sides of an angle is the bisector of the angle.
(11.5)