GCD. In the two triangles GAB and GBC, we have GB common, AB = BC and GA= GC; hence the triangles are equal (B. I., T. XXV.), and being isosceles since GA =GB=GC, we have the angles GAB=GBA=GBC=GCB. That is, each of these angles is one half the angle of the polygon; hence the angle GCB=GCD. Now, comparing the two triangles GBC and GCD, we have the side GC common, BC= CD, and the angle GCB=GCD; these triangles are therefore equal (B. I., T. XX.); and we have GD = GB=GC=GA; hence, the circumference which passes through A, B, and C, will also pass through D. In a similar manner we can show that the circumference which passes through B, C, and D, will also pass through E, and so on, for all the angles of the polygon, which demonstrates the first part of the Theorem. As to the second part, we observe, that the sides AB, BC, CD, etc., are equal chords, in reference to the circumscribed circle, and therefore they are equally distant from the centre G (T. VII., C. I.). If then, with G as a centre, and with the perpendicular GH as a radius, we describe a circumference, it will be tangent to all the sides of the polygon. This circle will be inscribed in the polygon, and the polygon will circumscribe the circle. Scholium. The point F, the common centre of the inscribed and circumscribed circles, is called the centre of the regular polygon. The radius of the circumscribing circle is called the radius of the polygon, and the radius of the inscribed circle is called. the apothem. The angles AGB, BGC, CGD, etc., are called angles at the centre. The lines GA, GB, GC, etc., which are the radii of the polygon, bisect the angles of the polygon. The apothems GH, GK, GL, etc., which are the radii of the inscribed circle, bisect the sides of the polygon perpendicularly, and also bisect the angles at the centre, since they would, if produced, bisect the arcs which measure the angles at the centre. 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. A" E' A' B 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 angles 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.). M D 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 с 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 |