cuts them, the angle ACE is equal to DEC. The triangles ACE and DEC have the side CE common, the side AC equal to ED, and the included angle ACE equal to the included angle DEC; they are therefore equal in all their parts (P. 8); consequently AE is equal to CD. Hence, the opposite sides of ACDE are equal each to each, and the figure is a parallelogram (P. 27), which was to be proved.

Cor. If two points on the same side of a line are equally distant from that line, the line that joins them is parallel to the given line.

PROPOSITION XXIX, THEOREM.

The diagonals of a parallelogram mutually bisect each other.

Let ACDE be a parallelogram whose E diagonals intersect at O. The triangles ACO and DEO, have the side AC equal to

DE, the angle ACO equal to DEO, and the angle OAC equal to ODE; they are therefore equal in all their parts. Hence, OA is equal to OD, and OC to OE, which was to be proved.

Cor. The diagonals of a rhombus are perpendicular to each other. For, if CD is equal to DE, the triangle CDE is isosceles, and because DO is drawn from the vertex D to the middle of the base CE, the line DA is perpendicular to CE.

EXERCISES ON BOOK I.

Let the student demonstrate the following theorems:

1o. If the sum of two adjacent angles is equal to two right angles, the bisectrices of these angles are perpendicular to each other.

2°. The bisectrices of two opposite angles of a parallelogram are parallel to each other.

3°. The bisectrices of two consecutive angles of a parallelogram are perpendicular to each other.

4°. The middle points of the sides of any quadrilateral are the vertices of a parallelogram.

5°. If two lines mutually bisect each other, their extremities are the vertices of a parallelogram.

6°. If the opposite angles of a quadrilateral are equal, each to each, the quadrilateral is a parallelogram.

yo. If the diagonals of a parallelogram are equal, the parallelogram is a rectangle.

8°. The diagonals of a rhombus are the bisectrices of the angles of the rhombus.

9°. If a line included between two parallels is bisected and if a second line is drawn through the point of bisection and limited by the parallels, then will the second line be bisected at the same point.

10°. The lines that join the middle points of the sides of any triangle divide that triangle into four equal parts.

BOOK II.

THE CIRCLE, AND THE MEASURE OF ANGLES.

Definitions.

47. A Circle is a portion of a plane bounded by a curve all of whose points are equally distant from a point within, called the centre. The bounding line is called the circumference.

The centre of a circle is also the centre of its circumference.

48. A radius is a line drawn from the centre to any point of the circumference, as OA.

All radii of the same circle, or of equal circles, are equal.

D

E

A

49. A diameter is a line drawn through the centre and limited by the circumference, as AE.

K

A diameter is double the radius; all diameters of the same circle, or of equal circles, are equal.

50. An arc is a part of a circumference, as ACD. The line that joins the extremities of an arc is called a chord; thus, AD is the chord of the arc ACD and also of the arc AKD.

A chord is said to subtend the arc to which it belongs; thus, AD subtends the arc ACD and also the arc AKD.

51. An angle at the centre is an angle formed by two radii, as AOD.

The intercepted arc ACD is said to subtend the angle AOD.

52. A sector is a part of a circle bounded by two radii and their intercepted arc, as AODC.

C

E

A

53. A segment is a part of a circle bounded by an arc and its chord, as ADC.

K

54. Similar arcs, similar sectors, and similar segments, are those which correspond to equal angles at the

centre.

Two arcs cannot be made to coincide unless they belong to the same circumference, or to equal circumferences.

A circle, or a circumference, may be designated by two letters, one at the centre and the other at the extremity of any radius; thus, the circle whose centre is O and whose circumference is ACK is called the circle OA.

Any diameter divides the circle to which it belongs into two equal parts.

Let AE be any diameter of the circle OA: then are the parts ACE and AKE equal to each other.

K

Let the part of the plane above AE be revolved about that line till it falls on the part below; each point of the curve ACE will fall on some point of the curve AKE, because all the points of both are equally distant from 0; hence, the two parts of the circle coincide throughout; they are therefore equal, which was to be proved.

Cor. Any diameter bisects a circumference.

Scho. The segments formed by a diameter are semi-circles, and the arcs subtended by a diameter are semi-circumferences.

PROPOSITION II. THEOREM.

A diameter is greater than any other chord.

Let AD be any chord, not a diameter, of the circle OA, and let AE be a diameter through A: then is AE greater than AD.

Draw the radius OD.

E

A

K

The sum of the radii OA and OD is greater than AD (P. 6, B. 1); but the sum of OA and OD is equal to AE; hence, AE is greater than AD, which was to be proved.

Scho. The segments ACD and AKD are unequal; the arcs ACD and AKD, subtended by AD, are also unequal.

PROPOSITION III. THEOREM.

Equal angles at the centres of equal circles are subtended by equal arcs; and conversely, equal arcs subtend equal angles at the centre.

on the sector AOD so that the angle Q shall coincide with the angle O; the point E will fall on A and the point G on D, and since all the points of the arcs EFG and ACD are