THEOREM XVI.

If through the corners of a regular polygon, already inscribed in a circle, we draw tangents, we shall thus circumscribe a regular polygon of the same number of sides.

The tangents thus drawn will form with the sides of the inscribed polygon, AB, BC, CD, etc., a series of triangles, AA'B, BB'C, CC'D, etc., all isosceles and equal, since AB BC= CD etc., and the an

=

=

E'

E"

B'

B

D'

DI

gles A'AB, A'BA, B'BC, B'CB, C'CD, C'DC, etc., are all equal, having for their measure half of the equal arcs AB, BC, CD, etc. (T. X., C. II.). Hence, all the angles, A', B', C', etc., of the circumscribed polygon are equal. And since AA' A'B = BB'B'C=CC' = etc., we have A'B'=B'C' C'D' etc. That is, this circumscribed polygon has all its angles equal, and all its sides equal; it is therefore regular (D. VI.).

=

Scholium. Since the radius OA' of the circumscribed polygon bisects the angle E'A'B' (T. XV., S.), we have, in the two rightangled triangles OAA' and OBA', the acute angle OA'A of the one, equal to the acute angle OA'B of the other; consequently, the remaining acute angles are equal; that is, the angle A'OA = A'OB. In the same way we can show that the angle B'OB = B'OC. But the angle A'OB = B'OB, since the apothem FB bisects the angle A'FB' at the centre (T. XV., S.). Hence, all the angles AOA', A'OB, BOB', B'OC, COC', etc., are equal; and their measuring arcs AI, IB, BK, KC, CL, etc., are all equal.

If then we suppose the circumscribed polygon to revolve in its own plane about its centre O, as a pivot, so that A may reach the point I, B will then coincide with K, C with L, etc. The circumscribed polygon will by this means, without having its relative parts in the least changed, assume a new position, having its sides parallel with the sides of the inscribed polygon.

OF SECANT AND TANGENT CIRCLES.

THEOREM XVII.

If two circumferences of circles have two points in common, the line joining their centres will bisect their common chord perpendicularly.

For, the line bisecting a chord at right angles passes through the centre (T. IV., S.); and as this chord is common to both circles, this bisecting line must pass through both centres.

THEOREM XVIII.

When two circumferences have only one point in common, this point will be situated on the line joining their centres.

A

C

E

B

D

If A and B are the centres of two circles having only one point in common, that point will be situated on the line AB joining their centres. For, if not, suppose it to be situated without this line, as at C. Draw CE perpendicular to AB, and prolong it until ED = CE; then we evidently have AC AD, also BC= BD, and since C was a common point of the circumferences, D will also be a common point. These circumferences will then have two points in common, which is contrary to the hypothesis. Hence, the point in common to the two circumferences must be in the line joining their

centres.

=

Scholium. Suppose we consider two circles whose centres are at A and B, the one at B being the smaller. Draw CF passing through their centres, and at the extremities of their diameters, CD and EF, draw the perpendiculars GG', HH', KK', LL'. If we suppose the circle, whose centre is at A, fixed, and the other to move towards it, so that the centre B may move in the line AB, we may notice five distinct positions of these circles.

Fourthly. When the tangent LL' coincides with HH', and becomes a common tangent to the two circumferences. In this case, D is the only point common to the two circumferences. For, drawing any radius, AM, of the larger circle, cutting the circumference of the smaller circle at N, and we shall have

AN<AB+BN, or AN<AB+BD, or AN<AD, or AM.

H

Hence, every point in the circumference of the smaller circle, except the point D, is within the circumference of the larger circle. The circles are said to touch internally.

Fifthly. When the limiting tangents of the smaller circle are both situated between those of the larger circle. In this case the two circumferences have no common point, and the smaller is wholly within the larger. For, joining any point of its circumference, as N, with the centres, we have

AN <AB+BN, or AN <AB+BF, but AB+BF=AF, which is less than AD; hence, AN <AD.

THEOREM XIX.

When two circumferences have no point in common, the dis tance between their centres is either greater than the sum of their radii, or less than their difference.

For by the Scholium under the last Theorem we have two positions of the circumferences in which they have no common point. First. When they are, the one wholly without the other, in which case the distance between their centres is greater than the sum of their radii.

Secondly. When the smaller circumference lies wholly within the larger; in which case the distance between their centres is less than the difference of their radii.

If we denote the greater radius by R, the less radius by R', and the distance between their centres by D, we shall have, when the circles are exterior,

D>R+R',

and when they are interior-that is, when the one is wholly within the other-we shall have

D<R-R'.

THEOREM XX.

When two circumferences are tangent to each other, the distance between their centres is equal to the sum of their radii, or equal to their difference.

There are two positions of the circumference in which

they are tangent to each other (T. XVIII., S.). In the first

case,

D=R+R',

and in the second case,

D=R-R'.

THEOREM XXI.

When two circumferences cut each other, the distance between their centres is less than the sum of their radii, and greater than their difference.

By the third case under the Scholium of Theorem XVIII. (see figure of the same), we see that the point M, common to the two circumferences, must be necessarily without the line AB which joins their centres; hence the three points A, B, and M must form a triangle, consequently we must have

D<R+R' and D>R- R'.

Scholium. The reciprocals of the foregoing Theorems may readily be demonstrated by the method of reducing to an absurdity. They may be thus stated:

When two circumferences are in the same plane:

1. If D>R+R', the two circles are exterior the one to the other.

2. If D=R+R', the two circles will touch externally.

3. If DR+R', and at the same time D> RR', the two circles will cut each other in two points.

4. If D=R-R', the two circles will touch internally.

5. If D <R - R', the smaller circle will lie wholly within the other.