THEOREM XXXIX.

If two circles touch one another the line joining their centres passes through the point of contact.

Let B be the centre of a circle touching the circle A internally, and C the centre of a circle touching the circle A externally; then shall the straight lines A B and AC pass through the point of contact, D. Join A D, CD, B D.

B

Since AD and CD are both at right angles to the common tangent DT, at D, therefore they are in the same straight line.

Similarly, BD and AD, being both perpendicular to DT at the point D, are therefore in the same straight line.

THEOREMS FOR EXERCISE.

79. How many equal circles will exactly circumscribe a given circle of the same radius ?

80. Show that all equal straight lines in a circle will be touched by another circle.

81. If from two fixed points in the circumference of a circle straight lines be drawn, intercepting a given arc and meeting without the circle, the locus of their intersections is a circle.

82. Two circles intersect in two points A and B: find the position of the longest straight line drawn through A and terminated by the two circumferences.

83. If through the point of contact of two circles which touch one another straight lines be drawn and terminated by the circumferences, the lines joining their extremities shall be parallel.

CHAPTER X.

POLYGONS.

Definition. A plane figure bounded by straight lines is called a polygon. Some polygons have special names.

A polygon of three sides is termed a triangle.

quadrilateral.
pentagon.
hexagon.

heptagon. octagon. decagon. dodecagon.

It is immaterial whether we name the polygon from the number of its sides or the number of its angles; for it is evident that every polygon has as many sides as angles. For instance, a quadrilateral (a foursided figure) is also called a tetragon (i.e. a figure with four angles).

Important General Properties. We will proceed to notice some of the most important general properties of polygons.

1. If lines be drawn from any one of the angular points of a polygon to all the others, the polygon is divided into as many triangles, less two, as it has sides.

For example, let the polygon have seven sides (Fig. 117). If one apex be joined to all the others, we have six lines, which form between them, at the apex (7-2) angles, and divide the polygon into (7-2) corresponding triangles. Similarly, if the polygon have n sides, it may be divided into n-2 triangles by diagonals from the same apex. There will be the same number of triangles if the figure be divided by diagonals drawn in any way so as not to intersect within the figure. (Fig. 117.)

2. The number of diagonals of a polygon of n sides is n (n—3).

If the polygon have, for instance, seven sides, then from one apex (7-3) diagonals are drawn, and so from all the angular points seven times this number may be drawn. It will be seen, however, that since each diagonal connects two angles, it will be drawn twice over; and therefore, to obtain the number of possible diagonals, it will be necessary to divide

FIG. 117

FIG. 116.

the preceding product by two. Thus, in a polygon of seven sides there are 7 x 4, or fourteen diagonals. Again, in a pentagon we have5 × 2

2

or five diagonals. In a polygon of n sides § n (n—3) diagonals.

2

3. The sum of the interior angles of a polygon is equal to as many times two right angles as there are sides, less two.

Since we can divide the polygon into as many triangles as there are sides, less two, and since the sum of the angles of each triangle is equal to two right angles, the angles of all the triangles will be equal to as many times two right angles as there are sides, less two. As the sum of the angles in all the triangles is equal to the sum of the angles of the polygon, the latter sum will be as many times two right angles as there are sides, less two.

The sum of the angles of a heptagon will be (7—2), or five times two right angles; of a decagon eight times two right angles, &c.

Regular Polygons.-Polygons which have their sides equal and their angles equal are called regular polygons. The sides of a figure may be equal and the angles unequal, but such a figure will not be regular. Take a folding pocket-measure, formed of a number of equal rules jointed together, and join the two ends; a polygon will be formed, the sides of which are equal (Fig. 118), but it will not be regular

FIG. 118.

unless it be also equi-angular.

FIG. 119.

If the two conditions be fulfilled,

the figure will be a regular polygon (Fig. 119).

Relation to the Circle.-The two most prominent features of a regular polygon are, first, its roundness, and, second, its symmetry about certain lines through a point within it which may be regarded as the centre. A regular polygon and a circle are thus intimately related. If a circle be made to pass through three angular points of a regular polygon, it will pass through all; if a circle be made to touch three of the sides, it will touch all. If a circumference be divided into equal parts, and the points of division joined, or if tangents be drawn through them, a regular polygon will be formed.

Symmetrical Properties of Polygons.—The following are the principal symmetrical properties of regular polygons :—

1. The diagonals which join the opposite vertices in a regular polygon of an even number of sides are diameters of the circumscribing

circle.

These vertices (Fig. 120) contain between them the same number of sides; moreover the sides, and consequently the subtended arcs, are all equal; hence the sum of these arcs in one direction is equal to the sum of the arcs in the other direction: and therefore the vertices bisect the circumference, and are at the opposite extremities of a diameter.

The point of intersection of two diagonal diameters is the centre of a regular polygon of an even number of sides.

FIG 120.

2. The straight line which joins the middle points of two opposite sides in a regular polygon of an even number of sides is a diameter of the inscribed circle.

For this straight line joins the points of contact of two parallel tangents to the inscribed circle, and consequently

must pass through its centre.

3. If the number of sides in the regular polygon be odd, the perpendicular let fall from an apex to the side opposite to it passes through the centre of the inscribed and circumscribed circles (Fig. 121), and is a symmetrical axis.

This may be shown in the same way as Proposition I.

FIG. 121.

4. Every regular polygon has as many symmetrical axes as sides. For, if the figure have an even number of sides, eight for example, there are four diagonals which are symmetrical axes, and four diameters, perpendicular to the sides, which are also symmetrical axes. If the number of sides be odd, seven for instance, the perpendiculars let fall from the seven vertices upon the sides opposite to them are symmetrical axes.

5. A regular polygon with an even number of sides has a centre of symmetry which is the common centre of the circumscribed and inscribed circles.

A centre of symmetry is a point such that all lines drawn through it to the boundary of the figure are bisected at that point.

This is the case with a regular polygon of an even number of sides ;

for any line through the centre terminates either in the circumference or in parallel and equal chords.

6. The angles at the centre of a regular polygon are equal to one

another.

Let the circumscribing circle be drawn, then these angles stand upon arcs subtended by equal chords.

7. If through the angles of a regular inscribed polygon tangents be drawn to the circle, they form a regular circumscribed polygon of the same number of sides. (Fig. 122.)

For the figure will be symmetrical about the line drawn through the point of contact of any tangent and the centre; and therefore all the sides and all the angles of the circumscribed polygon will be equal.

8. If tangents be drawn parallel to the sides of a regular inscribed polygon, they form a regular circumscribed polygon of the same number of sides. (Fig. 123.)

For a diameter drawn from any point of contact will be perpendicula to the tangent, and, therefore, also to the side of the polygon parallel to it. Consequently, the figure will be symmetrical about this diameter, and the lines and angles on one side will be equal to the corresponding lines and angles on the other.

Hence, whenever we know how to inscribe a regular polygon in a circle, we are able also to describe about the same circle a regular polygon of the same number of sides.

9. The angle of a regular polygon of n sides is equal to two right angles, minus the nth part of four right angles.

The number of degrees in the angle at the centre of a regular polygon will be obtained by dividing 360 by the number of sides in the polygon; and the interior angle, that is the angle formed by two adjacent sides of the polygon, is equal to the sum of the equal angles of the isosceles triangle formed by drawing radii to the extremities of a side, and is therefore 180°, minus the angle at the centre of the polygon.

Thus the angle at the centre of a polygon of nine sides is 40°, and the angle of the polygon 140°; of a polygon of seven sides the angle at