Draw the straight line DD' bisecting AB at right angles.

and EE' bisecting BC at right angles.

Prob. 4.

Prob. 4.

Then because every point equidistant from A and B lies on DD', Locus iii. and every point equidistant from B and C lies on EE', Locus iii. therefore any point equidistant from A, B and C must lie on DD' and on EE'.

Now DD' and EE' intersect, since if they were parallel, AB and BC, which are perpendicular to them, would lie in one straight line;

let them intersect in O,

then a point O has been found equidistant from A, B and C.

Also, because DD' and EE' can intersect in one point only,

therefore O is the only point equidistant from A, B and C.

Ax. 2.

ii. To find a point equidistant from three given straight lines which intersect one another, but not in the same point.

Let AA', BB', CC' be three given straight lines which intersect so as to form a triangle DEF :

it is required to find a point equidistant from AA', BB', and CC'.

Prob. I.

Draw the pair of straight lines MM', NN' bisecting the angles formed by AA' and BB', and the pair of straight lines PP', QQ', bisecting the angles formed by BB' and CC'.

Prob. I.

Then because every point equidistant from AA' and BB' lies on MM' or NN',

Locus iv.

and every point equidistant from BB' and CC'lies on PP' or QQ',

Locus iv.

therefore any point equidistant from AA', BB' and CC' must lie on MM' or NN', and also on PP' or QQ'.

Now MM' and PP' are not parallel,

since the angles MFD, PDF, being half the angles EFD, EDF of the triangle DFF, are not together equal to two right angles; also MM' and QQ' are not parallel,

since the angle MFD, being half the angle EFD, is not equal to

the angle QDB', which is half the exterior angle EDB' of the triangle DEF ;

let MM' intersect PP' in O, and QQ' in O.

Similarly let NN' intersect PP' in O1, and QQ' in O„.

1:

3

Then four points O, O1. O2, O, have been found equidistant from AA', BB', and CC'.

only,

Also, because two straight lines can intersect in one point

3

Ax. 2.

therefore O, O,, O2, O, are the only points equidistant from AA', BB', and CC'.

EXERCISES.

107. Find the points in a given straight line which are at a given distance from a given point.

108. In a given straight line find points at a given distance from another given straight line.

109. In a given straight line find a point equidistant from two given points.

110. Three unlimited straight lines form a triangle: find points

in one of them which are equidistant from the other two. 111. How many points are there in a plane each of which is equidistant from two given unlimited straight lines, as well as from two given points situated in that plane?

DEFINITIONS, AXIOMS, AND POSTULATES

BOOK I.

OF

DEFINITIONS.

DEF. 1. A point has position, but it has no magnitude.

DEF. 2. A line has position, and it has length, but neither breadth nor thickness.

The extremities of a line are points, and the intersection

of two lines is a point.

DEF. 3. A surface has position, and it has length and breadth, but not thickness.

The boundaries of a surface, and the intersection of two surfaces, are lines.

DEF. 4. A solid has position, and it has length, breadth and thick

ness.

The boundaries of a solid are surfaces.

DEF. 5. A straight line is such that any part will, however placed, lie wholly on any other part, if its extremities are made to fall on that other part.

DEF. 6. A plane surface, or plane, is a surface in which any two points being taken the straight line that joins them lies wholly in that surface.

DEF. 7. When two straight lines are drawn from the same point, they are said to contain, or to make with each other, a plane angle. The point is called the vertex, and the straight lines are called the arms, of the angle.

A line drawn from the vertex and turning about the

vertex in the plane of the angle from the position of coincidence with one arm to that of coincidence with the other is said to turn through the angle: and the angle is greater as the quantity of turning is greater. Since the line may turn from the one position to the other in either of two ways, two angles are formed by two straight lines drawn from a point. These angles (which have a common vertex and common arms) are said to be conjugate. The greater of the two is called the major conjugate, and the smaller the minor conjugate, angle.

When the angle contained by two lines is spoken of without qualification, the minor conjugate angle is to be understood. It is seldom requisite to consider major conjugate angles before Book III.

When the arms of an angle are in the same straight line, the conjugate angles are equal, and each is then said to be a straight angle.

DEF. 8. When three straight lines are drawn from a point, if one of them be regarded as lying between the other two, the angles which this one (the mean) makes with the other two (the extremes) are said to be adjacent angles: and the angle between the extremes, through which a line would turn in passing from one extreme through the mean to the other extreme, is the sum of the two adjacent angles.

DEF.

9.

The bisector of an angle is the straight line that divides it into two equal angles.

DEF. 10. When one straight line stands upon another straight line and makes the adjacent angles equal, each of the angles is called a right angle.