68. Problem: -To find the point that is equally distant from three given points, that are not in the same straight line.

Let A, B, C be the three given points.

It is required to find a point equally distant from A, B and C.

Draw EF the locus of all points that are equally distant from A and B. (I-22, p. 78.)

F

Draw GH the locus of all points that are equally distant from B and C.

Let EF and GH meet at K.

Then K is the required point.

K is on EF, . KA = KB.

K is on GH, .. KB = KC.

Consequently K is equally distant from A, B and C.

69.-Exercises

1. Find the locus of the centres of all circles that pass through two given points.

2. Describe a circle to pass through two given points and have its centre in a given st. line.

3. Describe a circle to pass through two given points and have its radius equal to a given st. line. Show that

generally two such circles may be described. there be only one? and when none?

When will

4. Find the locus of a point which is equidistant from two given || st. lines.

5. In a given st. line find two points each of which is equally distant from two given intersecting st. lines.

When will there be only one solution?

6. Find the locus of the vertices of all As on a given base which have the medians drawn to the base equal to a given st. line.

7. Find the locus of the vertices of all As on a given base which have one side equal to a given st. line.

8. Construct a ▲ having given the base, the median drawn to the base, and the length of one side.

9. Find the locus of the vertices of all As on a given base which have a given altitude.

10. Construct a ▲ having given the base, the median drawn to the base, and the altitude.

11. Construct a ▲ having given the base, the altitude and one side.

12. Find the locus of a point such that the sum of its distances from two given intersecting st. lines is equal to a given st. line.

13. Find the locus of a point such that the difference of its distances from two given intersecting st. lines is equal to a given st. line.

14. Find the locus of the vertices of all As on a given base which have the median drawn from one end of the base equal to a given st. line.

15. Show that, if the ends of a st. length slide along two st. lines at rt. the locus of its middle point is a circle.

cm. from AB.

line of constant

s to each other,

16. AB is a st. line and C is a point at a distance of 2 Find a point which is 1 cm. from AB and How many such points can be found?

4 cm. from C.

17. Two st. lines, AB, CD, intersect each other at an of 45°. Find all the points that are 3 cm. from AB and 2 cm. from CD.

18. ABC is a scalene A. Find a point equidistant from AB and AC, and also equidistant from B and C.

19. Find a point equidistant from the three vertices of a given A.

20. Find four points each of which is equidistant from the three sides of a A.

NOTE.-Produce each side in both directions.

21. Find the locus of a point at which two equal segments of a st. line subtend equals.

22. Find the locus of the centre of a circle which shall pass through a given point and have its radius equal to a given st. line.

23. A st. line of constant length remains always || to itself, while one of its extremities describes the circumference of a fixed circle. Find the locus of the other extremity.

24. The locus of the middle points of all st. lines drawn from a fixed point to the circumference of a fixed circle is a circle.

Miscellaneous Exercises

1. If a st. line be terminated by two ||s, all st. lines drawn through its middle point and terminated by the same s are bisected at that point.

2. If two lines intersecting at A be respectively to two lines intersecting at B, each at A is either equal to or supplementary to each L at B.

3. If two lines intersecting at A be respectively to two lines intersecting at B, each at A is either equal to or supplementary to each at B.

4. If from any point in the bisector of an st. lines be drawn to the arms of the L and

terminated by the arms, these st. lines are equal to each other.

X 5. In the base of a find a point such that the st. lines drawn from that point || to the sides of the ▲ and terminated by the sides are equal to each other.

6. One of an isosceles A is half each of the others. Calculate the s.

7. If the from the vertex of a A to the base falls within the ▲, the segment of the base adjacent to the greater side of the ▲ is the greater.

8. If a star-shaped figure be formed by producing the alternate sides of a polygon of n sides, the sum of the s

at the points of the star is (2

n

8) rt. s.

9. In a quadrilateral ABCD,

A

= B and C = D.

Prove that AD = BC.

10. The bisectors of the s of a gm form a rectangle, the diagonals of which are || to the sides of the original ||gm ; and equal to the difference between them.

11. From A, B the ends of a st. line is AC, BD are drawn to any st. line. E is the middle point of AB.

x12. If through a point within a three st. lines be drawn from the vertices to the opposite sides, the sum of these st. lines is greater than half the perimeter of the A.

13. A, D are the centres of two circles, and AB, DE are two || radii. EB cuts the circumferences again at C, F. Show that AC || DF.

X 14. The bisectors of the interiors of a quadrilateral form a quadrilateral of which the opposites are supplementary.

15. In a given square inscribe an equilateral ▲ having one vertex at a vertex of the square.

16. Through two given points draw two st. lines, forming an equilateral ▲ with a given st. line.

17. Draw an isosceles ▲ having its base in a given st. line, its altitude equal to a given st. line, and its equal sides passing through two given points.

18. If a be drawn from one end of the base of an isosceles to the opposite side, the between the

the base

=

half the vertical of the A.

and

19. If any point P in AD the bisector of the A of A ABC be joined to B and C, the difference between PB and PC is less than the difference between AB and AC.

20. If any point P in the bisector of the exterior - at A in the ABC be joined to B and C, PB + PC > AB +

AC.