139. Prove Theorem IX. by first drawing a line through B parallel to A C.

Why is not this method of proof adopted in (73)?

140. Prove Theorem IX. by drawing a triangle upon the floor, walking over its perimeter, and turning at each vertex through an angle equal to the angle at that vertex.

141. If a line joining two parallels is bisected, any other line drawn through the point of bisection and joining the parallels is bisected.

142. If on each side of a triangle an equiangular triangle is constructed externally to the triangle, the straight lines drawn from the remote vertices of the equilateral triangles to the opposite angles of the given triangle are equal.

143. The three perpendiculars from the vertices of a triangle to the opposite sides intersect at the same point.

Through the vertices draw lines parallel to the opposite sides forming a second triangle. (117; 107.)

144. A line from the vertex of the right angle of a right triangle bisecting the hypothenuse is equal to half of the hypothenuse.

Let A B C be the triangle right-angled at C, and from C adjacent to the side B C cut off an angle equal to B. (77; 85.)

145. If one of the acute angles of a right triangle is double the other, the hypothenuse is double the shortest side. (144; 82; 86.)

146. Prove in Theorem XV. the angles of the two triangles equal by reference to (102); then that the triangles are equal by (80) or (81).

147. (Converse of part of 117.) If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.

148. (Converse of 118.) If a diagonal divides a quadrilateral into two equal triangles, is the figure necessarily a parallelogram?

149. (Converse of 122.) If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.

150. (Converse of 123.) If the diagonals of a quadrilateral bisect each other at right angles, the figure is a rhombus or a square.

2

151. The diagonals of a rectangle are equal.

152. The diagonals of a rhombus bisect the angles of the rhombus.

153. Straight lines bisecting the adjacent angles of a parallelogram are perpendicular to each other.

154. From the vertices of a parallelogram measure equal distances upon the sides in order. The lines joining these points on the sides form a parallelogram.

155. Prove Theorem XXIX. by joining any point within to the vertices of the polygon.

156. If the sides of a polygon, as ABCDEF, are produced, the sum of A the angles a, b, c, d, e, f, is equal to four right angles.

B

C

le

Prove by reference to (125) and (44). Also by drawing from any point lines parallel to the several sides forming the exterior angles, and referring to (46) and (49).

157. If a pavement is to be laid with blocks of the same regular form, that is, blocks whose faces are equiangular and equilateral, prove that their upper faces must be equilateral triangles, squares, or hexagons. (125; 46.)

158. If two kinds of regular figures, with sides of the same length, are to be used at each angular point, show that the pavement can be laid only with blocks whose upper faces are,

How many of each must there be at each angular point?

159. If three kinds of regular figures, with sides of the same length, are to be used at each angular point, show that the pavement can be laid only with blocks whose upper faces are,

1st. Triangles, squares, and hexagons.

2d. Squares, hexagons, and dodecagons.

How many of each must there be at each angular point?

BOOK II.

RELATIONS OF POLYGONS.

RATIO AND PROPORTION.

[As it is necessary to understand the elementary principles of ratio and proportion before entering upon the Books that are to follow, it is introduced here, though it belongs properly to Algebra.]

DEFINITIONS.

1. Ratio is the relation of one quantity to another of the same kind; or it is the quotient which arises from dividing one quantity by another of the same kind.

Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to b is written,

α

a : b, or 7; read, a is to b, or a divided by b.

2. The Terms of a ratio are the quantities compared, whether simple or compound.

The first term of a ratio is called the antecedent, the other the consequent; the two terms together are called a couplet.

3. An Inverse or Reciprocal Ratio of any two quantities is the ratio of their reciprocals. Thus, the direct ratio of a to b

α

1 1 is ab, that is, ; the inverse ratio of a to b is : that is, b'

a

4. Two quantities are commensurable if there is a third quantity of the same kind which is contained an exact number of times in each. This third quantity is called the common measure of these two quantities.

ALL

В ШШ

CU

Thus, the two lines A and B are commensurable if there is a third line C which is contained an exact number of times in each, as for example, 9 times in A, and 5 times in B; and the third line C is the common measure of A and B. ratio of two commensurable quantities therefore can always be exactly expressed in numbers. The ratio of A to B is.

The

9

Two quantities are incommensurable if they have no common

measure.

The ratio of two quantities, as A and B, whether commenA surable or not, is expressed by If A and B are incommen

A

Β ́

surable, is called an incommensurable ratio.

A constant ratio is a ratio which remains the same though its terms may vary. Thus, the ratio of 3:4, 6:8, 9:12, is constant; also the ratio of A: B and m Am B.

5. Proportion is an equality of ratios. Four quantities are in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth.

The equality of two ratios is indicated by the sign of equality (=), or by four dots (: :).

α

C

b d

Thus, abc: d, or a : b::c:d, or = read a to b equals c to d, or a is to b as c is to d, or a divided by b equals c divided by d.

In a proportion the antecedents and consequents of the two ratios are respectively the antecedents and consequents of the proportion. The first and fourth terms are called the extremes, and the second and third the means.

6. When three quantities are in proportion, e. g. a:bb: c, the second is called a mean proportional between the other two; and the third, a third proportional to the first and second.

7. A proportion is transformed by Alternation when antecedent is compared with antecedent, and consequent with consequent.

8. A proportion is transformed by Inversion when the antecedents are made consequents, and the consequents antecedents.

9. A proportion is transformed by Composition when in each couplet the sum of the antecedent and consequent is compared with the antecedent or with the consequent.

10. A proportion is transformed by Division when in each couplet the difference of the antecedent and consequent is compared with the antecedent or with the consequent.

11. Axiom. Two ratios respectively equal to a third are equal to each other.