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

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Prob. 1.

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. 1.

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 DEF, 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 Oz. Similarly let NN' intersect PP' in O,, and QQʻ in Oz. Then four points O, 01: 0.2, 0, have been found equidistant from AA', BB', and CC'.

Also, because two straight lines can intersect in one point only, therefore 0, 0, 0, 0, are the only points equidistant from AA', BB', and CC'.

Ax. 2.

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 OF

BOOK I.

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 with-
out 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 diviles

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.

DEF. 11. A perpendicular to a straight line is a straight line that

makes a right angle with it. DEF. 12. An acute angle is that which is less than a right angle. DEF. 13. An obtuse angle is that which is greater than one right

angle, but less than two right angles. DEF. 14. A reflex angle is a term sometimes used for a major con

jugate angle. DEF. 15. When the sum of two angles is a right angle, each is

called the complement of the other, or is said to be com

plementary to the other. DEF. 16. When the sum of two angles is two right angles, each

is called the supplement of the other, or is said to be

supplementary to the other. DEF. 17. The opposite angles made by two straight lines that inter

sect are called vertically opposite angles. DEF. 18. A plane figure is a portion of a plane surface inclosed by

a line or lines. DEF. 19. Figures that may be made by superposition to coincide

with one another are said to be identically equal ; or they

are said to be equal in all respects. DEF. 20. The area of a plane figure is the quantity of the plane

surface inclosed by its boundary. DEF. 21. A plane rectilineal figure is a portion of a plane surface

inclosed by straight lines. When there are more than

three inclosing straight lines the figure is called a polygon. Def. 22. A polygon is said to be convex when no one of its angles

is reflex. DEF. 23. A polygon is said to be regular when it is equilateral and

equiangular; that is, when its sides and angles are equal.

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