Fig. 44.

Fig. 44.

83. Scholium. If we would apply this proposition to polygons, which have any re-enteringt angles; each of these angles is to be considered as greater than two right angles. But, in order to avoid confusion we shall confine ourselves in future to those polygons, which have only saliant angles, and which may be called convex polygons. Every convex polygon is such, that a straight line, however drawn, cannot meet the perimeter in more than two points.

THEOREM.

84. The opposite sides of a parallelogram are equal, and the opposite angles also are equal.

Demonstration. Draw the diagonal BD (fig. 44); the two triangles ADB, DBC, have the side BD common; moreover, on account of the parallels AD, BC, the angle ADB = DBC (67), and on account of the parallels AB, CD, the angle ABD=BDC; therefore the two triangles ADB, DBC, are equal (38); consequently the side AB opposite to ADB is equal to the side DC opposite to the equal angle DBC, and likewise the third side AD is equal to the third side BC; therefore the opposite sides of a parallelogram are equal.

Again, from the equality of the same triangles it follows, that the angle AC, and also that the angle ADC, composed of the two angles ADB, BDC, is equal to the angle ABC, composed of the two angles DBC, ABD; therefore the opposite angles of a parallelogram are equal.

85. Corollary. Hence two parallels AB, CD, comprehended between two other parallels AD, BC, are equal.

THEOREM.

86. If, in a quadrilateral ABCD (fig. 44), the opposite sides are equal, namely, AB = CD, and AD CB, the equal sides will be parallel, and the figure will be a parallelogram.

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Demonstration. Draw the diagonal BD; the two triangles ABD, BDC, have the three sides of the one equal to the three

† A re-entering angle is one whose vertex is directed inward, as Fig. 43. CDE (fig. 43), while a saliant angle has its vertex directed outward

as ABC.

sides of the other, each to each, they are therefore equal, and the angle ADB opposite to the side AB is equal to the angle DBC opposite to the side CD; consequently the side AD is parallel to BC (67). For a similar reason AB is parallel to CD; therefore the quadrilateral ABCD is a parallelogram.

THEOREM.

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87. If two opposite sides AB, CD (fig. 44), of a quadrilateral Fig. 44. are equal and parallel, the two other sides will also be equal and parallel, and the figure ABCD will be a parallelogram.

Demonstration. Let the diagonal BD be drawn; since AB is parallel to CD, the alternate angles ABD, BDC, are equal (67). Besides, the side AB = CD, and the side DB is common, therefore the triangle ABD is equal to the triangle DBC (36), and the side AD = BC, the angle ADB = DBC, and consequently AD is parallel to BC; therefore the figure ABCD is a parallel

ogram.

THEOREM.

88. The two diagonals AC, DB (fig. 45), of a parallelogram Fig. 45. mutually bisect each other.

Demonstration. If we compare the triangle ADO with the triangle COB, we find the side AD = CB, and the angle

also the angle DAO OCB; therefore these two triangles are equal (38), and consequently AO, the side opposite to the angle ADO, is equal to OC, the side opposite to the angle OBC; DO likewise is equal to OB.

89. Scholium. In the case of the rhombus, the sides AB, BC, being equal, the triangles AOB, OBC, have the three sides of the one equal to the three sides of the other, each to each, and are consequently equal; whence it follows, that the angle

AOB = BOC,

and that thus the two diagonals of a rhombus cut each other mutually at right angles.

Fig 46.

SECTION SECOND.

Of the Circle and the Measure of Angles.

DEFINITIONS.

90. THE circumference of a circle is a curved line all the points of which are equally distant from a point within called the centre. The circle is the space terminated by this curved line*.

91. Every straight line CA, CE, CD (fig. 46), &c. drawn from the centre to the circumference is called a radius or semidiameter, and every straight line, as AB, which passes through the centre and is terminated each way by the circumference, is called a diameter.

By the definition of a circle the radii are all equal, and all the diameters also are equal and double of the radius.

92. An arc of a circle is any portion of its circumference, as FHG.

The chord or subtense of an arc is the straight line FG, which joins its extremities**.

93. A segment of a circle is the portion comprehended between an arc and its chord.

94. A sector is the part of a circle comprehended between an arc DE and the two radii CD, CE, drawn to the extremities of this arc.

95. A straight line is said to be inscribed in a circle, when its Fig. 47. extremities are in the circumference of the circle, as AB (fig. 47). An inscribed angle is one whose vertex is in the circumference, and which is formed by two chords, as BAC.

An inscribed triangle is a triangle whose three angles have their vertices in the circumference of the circle, as BAC.

* In common discourse the circle is sometimes confounded with its circumference; but it will always be easy to preserve the exactness of these expressions by recollecting that the circle is a surface which has length and breadth, while the circumference is only a line.

** The same chord, as FG, corresponds to two arcs, and consequently to two segments; but, in speaking of these, the smaller is always to be understood, when the contrary is not expressed.

And in general an inscribed figure is one, all whose angles have their vertices in the circumference of the circle. In this case, the circle is said to be circumscribed about the figure.

96. A secant is a line, which meets the circumference in two points, as AB (fig. 48).

97. A tangent is a line which has only one point in common with the circumference, as CD.

The common point M is called the point of contact.

Fig. 48

Also two circumferences are tangents to each other (fig. 59, 60), Fig. 59, when they have only one point common.

A polygon is said to be circumscribed about a circle, when all its sides are tangents to the circumference; and in this case the circle is said to be inscribed in the polygon.

60.

THEOREM.

98. Every diameter AB (fig. 49) bisects the circle and its cir- Fig. 49. cumference.

Demonstration. If the figure AEB be applied to AFB, so that the base AB may be common to both, the curved line AEB must fall exactly upon the curved line AFB; otherwise, there would be points in the one or the other unequally distant from the centre, which is contrary to the definition of a circle.

THEOREM.

99. Every chord is less than the diameter.

Demonstration. If the radii CA, CD (fig. 49), be drawn from Fig. 49. the centre to the extremities of the chord AD, we shall have the straight line AD< AC + CD, that is, AD < AB (91).

100. Corollary. Hence the greatest straight line that can be Inscribed in a circle is equal to its diameter.

THEOREM.

101. A straight line cannot meet the circumference of a circle in more than two points.

Demonstration. If it could meet it in three, these three points being equally distant from the centre, there might be three equal straight lines drawn from a given point to the same straight line. which is impossible (54).

Fig. 50.

Fig. 50.

THEOREM.

102. In the same circle, or in equal circles, equal arcs are subtended by equal chords, and conversely, equal chords subtend equal

arcs.

Demonstration. The radius AC (fig. 50) being equal to the radius EO, and the arc AMD equal to the arc ENG; the chord AD will be equal to the chord EG.

For, the diameter AB being equal to the diameter EF, the semicircle AMDB may be applied exactly to the semicircle ENGF, and then the curved line AMDB will coincide entirely with the curved line ENGF; but the portion AMD being supposed equal to the portion ENG, the point D will fall upon G; therefore the chord AD is equal to the chord EG.

Conversely, AC being supposed equal to EO, if the chord AD EG, the arc AMD will be equal to the arc ENG.

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For, if the radii CD, OG, be drawn, the two triangles ACD, EOG, will have the three sides of the one equal to the three sides of the other, each to each, namely, AC EO, CD=OG and AD EG; therefore these triangles are equal (43); hence the angle ACD=EOG. Now, if the semicircle ADB be placed upon EGF, because the angle ACD EOG, it is evident, that the radius CD will fall upon the radius OG, and the point D upon G, therefore the arc AMD is equal to the arc ENG.

THEOREM.

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103. In the same circle, or in equal circles, if the arc be less than half a circumference, the greater arc is subtended by the greater chord; and, conversely, the greater chord is subtended by the greater

arc.

Demonstration. Let the arc AH (fig. 50) be greater than AD, and let the chords AD and AH, and the radii CD, CH, be drawn. The two sides, AC, CH, of the triangle ACH, are equal to the two sides AC, CD, of the triangle ACD, and the angle ACH is greater than ACD; hence the third side AH is greater than the third side AD (42), therefore the greater arc is subtended by the greater chord.

Conversely, if the chord AH be greater than AD, it may be inferred from the same triangles that the angle ACH is greater than ACD, and that thus the arc AH is greater than AD.