41. A line is proved to be the locus of points fulfilling a given condition when two propositions are established -namely, either the following opposite propositions :

1. Every point on the line fulfils the given condition. 2. Every point not on the line does not fulfil the given condition.

Or converse theorems

1. Every point on the line fulfils the given condition. 2. Every point which fulfils the given condition is on the line.

Or the following opposite theorems :

1. Every point which fulfils the given condition is on the line.

2. Every point which does not fulfil the given condition is not on the line.

THEOREM XXI.

The locus of points which are equidistant from two fixed points is the perpendicular bisector of the line joining those points.

Let A B be the two points, and let C E be the straight line which bisects the straight line A B at right angles in the point D; then we shall prove that CE is the locus of points equidistant from A and B, by showing

I. That if a point be in C E it will be at the same distance from A and B.

Proof. Since M and N are points at the same distance from a and from b, the straight line MN is at right angles to a b.-(Theorem XXI.)

PROBLEM VI.

To construct a triangle with sides equal to three given straight lines.

Let a, b, and c be the given straight lines.

Draw a

straight line, BC, equal to one of the given sides, and, with one of the extremities as centre and another of the given sides as radius, describe a circle; and then, with the other extremity as centre

and the third side as radius, describe another circle cutting the first in the points A and A'.

A B C is the triangle required.

A B C is another triangle equal to it on the opposite side of the same base.

PROBLEM VII.

To draw a straight line through a given point parallel to a given straight line.

Let M be the given point, and A B the given straight line. From M draw M D to any point, D, in AB.

M

N

B

as centre, and the DN. With D as

With D as centre, and D M as radius, describe an arc, M C, cutting A B in C; and, with M same radius, describe an arc, centre, and the chord CM as

radius, describe an arc cutting DE in N. M N, and M N will be parallel to A B.

Draw the line

Proof. The construction is that of Problem II. for making 4 NMD < M DA; consequently, since these are alternate angles, M N and A B are parallel.

PROBLEMS FOR EXERCISE.

1. Through a given point to draw a straight line which shall make a given angle with a given straight line.

2. Find a point in a line which shall be equidistant from two given points.

3. Through a given point draw a straight line which shall make equal angles with two intersecting lines.

4. Find a point in a given straight line at the same distance from each of two other straight lines.

5. Draw a straight line of given length parallel to a given straight line and having its extremities on two other given straight lines.

6. Through a given point draw a straight line of given length so that its extremities are on two parallel straight lines.

7. From a given point to draw three straight lines of given lengths so that their extremities may lie in the same straight line and may intercept equal distances on this line.

8. Through a given point draw a straight line which shall be bisected in that point and shall have its extremities on two given straight lines which intersect.

9. Through a given point draw a straight line cutting three straight lines which are drawn from the same point so that the two parts intercepted by the three straight lines shall be equal.

10. Construct an isosceles triangle, having given the base and altitude.

II. Construct a right angled triangle, having given the hypothenuse and one of the acute angles.

12. Construct a right-angled triangle, having given the hypothenuse and one of the other sides.

13. Construct on a given base an isosceles triangle each of whose sides shall be double of the base.

14. Construct a triangle, having given the base, the sum of the sides, and one of the angles at the base.

15. Construct a triangle, having given the base, the difference of the sides, and one of the angles at the base.

16. Construct a triangle, having given the base, the difference of the sides, and the difference of the angles at the base.

17. Construct an isosceles triangle, having given the vertical angle in magnitude and position and the position of a point in the base.

18. Construct a triangle, having given a side, an angle adjacent to the side, and the length of the bisector of the angle.

19. Construct a triangle, having given the sum of two angles and the sides opposite to them.

20. Construct a parallelogram, having given the sides and a diagonal.

21. Construct a parallelogram, having given a side and two diagonals.

22. Construct a parallelogram, having given the diagonals and the angle between them.

23. Construct a triangle, having given two sides and the line joining the middle point of the third side with the apex of the opposite angle.

24. Construct a triangle, having given two sides and a line joining the middle point of one of them with the opposite angle.

SECTION II.-Further Remarks on the Problems.

Problems and Theorems Mutually Related. The construction in Problem VI. fails if two of the straight lines (b and c, for instance) are together less than the third; for in this case the two circles drawn will not intersect. This Problem involves the converse of Theorem XII.

By stating them as follows, they may be readily compared.
THEOREM XII.-Hypothesis.-"If A B C be a triangle,"

Conclusion-"Any two of its sides are together greater than the

third."

PROBLEM VI.-Hypothesis.—"If the three lines a, b, c be such that any two are together greater than the third,”

Conclusion "The three straight lines will form a triangle."

That there can be but one triangle on the same side of the base having sides equal respectively to the three given lines is proved by Theorem XIV.

The following Problems bear to Theorems V., VI., and XV. the same relations as the above Problem VI. bears to Theorem XIV.

To describe a Triangle, when two sides and the included angle are given.-Make an angle, A,

To describe a Triangle, when two angles and the included side are

given.-Draw a straight line, AB,

equal to the given side, and at its extremities make angles A and B equal respectively to the given angles. The sides of these angles form the triangle required (Fig. 82).

To describe a Triangle, when two

sides and an angle opposite one of

B

L

с

FIG. 82.

them are given.—Make an angle A, equal to the given angle; produce