47°. Theorem.-Four points determine at most six lines; and four lines determine at most six points.

Proof.-1. Let A, B, C, D be the four points. The groups of two are AB, AC, AD, BC, BD, and CD; or six in all.

Therefore six lines at most are determined.

The

2. Let L, M, N, K be the lines. groups of two that can be made are KL, KM, KN, LM, LN, and MN; or six in all. Therefore six points of intersection at

most are determined.

K

N

M

Cor. In the first case the six lines determined pass by threes through the four points. And in the second case the six points determined lie by threes upon the four lines.

This reciprocality of property is very suggestive, and in the higher Geometry is of special importance.

Ex. Show that 5 points determine at most 10 lines, and 5 lines determine at most 10 points. And that in the first case the lines pass by fours through each point; and in the latter, the points lie by fours on each line.

48°. Def.—A triangle is the figure formed by three lines and the determined points, or by three points and the determined lines.

The points are the vertices of the triangle, and the linesegments which have the points as end-points are the sides. The remaining portions of the determined lines are usually spoken of as the "sides produced." But in many cases generality requires us to extend the term

extending outwards as far as required, are the sides produced.

The triangle is distinctive in being the rectilinear figure for which a given number of lines determines the same number of points, or vice versa.

Hence when the three points, forming the vertices, are given, or when the three lines or line-segments forming the sides are given, the triangle is com

X

pletely given.

This is not the case with a rectilinear figure having any number of vertices other than three.

If the vertices be four in number, with the restriction that each vertex is determined by the intersection of two sides, any one of the figures in the margin will satisfy the conditions.

Hence the giving of the four vertices of such a figure is not sufficient to completely determine the figure.

49°. Def.-1. The angles ABC, BCA, CAB are the internal angles of the triangle, or simply the angles of the triangle.

2. The angle DCB, and others of like kind, are external angles of the triangle.

3. In relation to the external angle DCB, the angle BCA is the adjacent internal angle, while the angles CAB and ABC are opposite internal angles.

4. Any side of a triangle may be taken as its base, and then the angles at the extremities of the base are its basal angles, and the angle opposite the base is the vertical angle. The vertex of the vertical angle is the vertex of the triangle when spoken of in relation to the base.

50°. Notation.—The symbol is commonly used for the word triangle. In certain cases, which are always readily apprehended, it denotes the area of the triangle.

The angles of the triangle are denoted usually by the capital letters A, B, C, and the sides opposite by the corresponding small letters a, b, c.

51°. Def.-When two figures compared by superposition coincide in all their parts and become virtually but one figure they are said to be congruent.

Congruent figures are distinguishable from one another only by their position in space and are said to be identically equal.

Congruence is denoted by the algebraic symbol of identity, =; and this symbol placed between two figures capable of congruence denotes that the figures are congruent.

Closed figures, like triangles, admit of comparison in two ways. The first is as to their capability of perfect coincidence; when this is satisfied the figures are congruent. The second is as to the magnitude or extent of the portions of the plane enclosed by the figures. Equality in this respect is expressed by saying that the figures are equal.

When only one kind of comparison is possible, as is the case with line-segments and angles, the word equal is used.

CONGRUENCE AMONGST TRIANGLES.

52°. Theorem.-Two triangles are congruent when two sides and the included angle in the one are respectively equal to two sides and the included angle in the other.

If AB A'B' the triangles

BC=B'C' are congru

and LB LB' ent.

Proof.-Place ABC on A

B

C A

C

AA'B'C' so that B coincides with B', and BA lies along B'A'.

AC lies along A'C' ;

(24°, Cor. 2)

and .. and the As coinciding in all their parts are congruent. (51°)

q.e.d.

Cor. Since two congruent triangles can be made to coincide in all their parts, therefore

When two triangles have two sides and the included angle in the one respectively equal to two sides and the included angle in the other, all the parts in the one are respectively equal to the corresponding parts in the other.

53°. Theorem.-Every point upon the right bisector of a segment is equidistant from the end-points of the segment.

AB is a line-segment, and P is any point on its right bisector PC. Then PA=PB.

Proof. In the As APC and BPC,

Def. 1.-A triangle which has two sides equal to one an

other is an isosceles triangle.

Thus the triangle APB is isosceles.

The side AB, which is not one of the equal sides, is called the base.

Cor. 1. Since the ▲APC=^\BPC,

Hence the basal angles of an isosceles triangle are equal to one another.

Cor. 2. From (52°, Cor.), LAPC=2BPC;

Therefore the right bisector of the base of an isosceles triangle is the internal bisector of the vertical angle. And since these two bisectors are one and the same line the converse is true.

Def. 2. -A triangle in which all the sides are equal to one another is an equilateral triangle.

Cor. 3. Since an equilateral triangle is isosceles with respect to each side as base, all the angles of an equilateral triangle are equal to one another; or, an equilateral triangle is equiangular.

54. Theorem.--Every point equidistant from the endpoints of a line-segment is on the right bisector of that segment. (Converse of 53°.)

PA=PB. Then P is on the right bisector

of AB.

Proof. If P is not on the right bisector of AB, let the right bisector cut AP in Q.

Therefore the right bisector of AB does not cut AP; and similarly it does not cut BP; therefore it passes through P, or P is on the right bisector.

q.e.d.

This form of proof should be compared with that of Art. 53°, they being the kinds indicated in 6°.

This latter or indirect form is known as proof by reductio ad absurdum (leading to an absurdity). In it we prove the conclusion of the theorem to be true by showing that the acceptance of any other conclusion leads us to some relation which is absurd or untrue.

55°. Def.-The line-segment from a vertex of a triangle to the middle of the opposite side is a median of the triangle. Cor. 1. Every triangle has three medians.

Cor. 2. The median to the base of an isosceles triangle is