GEOM BOOKS 1I.—1v.] This theorem may also be stated thus :— Through three points only one circumference may be drawn; or, T h rcc points determine a circle. Euclid does not give the theorem in this form. He proves, how- ever, that the two circles cannot cut (mother in more than two points (Prop. 10), and that too circles cannot touch one anothcr in more points tha.n onc (Prop. 13). _ 5‘ 30. Propositions 11 and 12 assert that if two czrclcs touch, then the point of contact lies on the line joining their centr_cs. This gives two 131-opositions, because the _circles may touch either internally or externally. _ § 31. Propositions 14 and 15 relate to the length of chords. The first says ; that cqual chords are equidistant from the centrc,ancl that c/mrols a_'hich are equidistant from the centre are equal ; Whilst Prop. 15 compares unequal chords, viz., Of all chords the diameter is the grcatcst, and of other chords that is the greater which is nearer to the centre ; and conversely, the grcatcr chord is ncarcr to the centre. 3' 32. In Prop. 16 the ta11gcnt to a circle is for the first time in- troduced. The proposition is meant to show that the straight line at the end point of the diameter, and at right angles to it is a. tan- gent. The proposition itself does not state this. It runs thus :— Prop. 16. The straight line (lI‘(tlL‘)t at right angles to the diameter of a circle, from the c.rtrcmiI_:/ of it, falls icithout the circle ; anrl no .s-traight line can be drawn from the castrcmitg, between that straight line and the circumference, so as not to cut the circle. Corollary/.—'l‘lie straight line at right angles to a. diameter drawn through the end point of it touches the circle. The statem-.-nt of the proposition and its whole treatment show the diflicultics which the tangents presented to Euclid. Prop. 17 solves the problem through a gircn point, cithcr in the circumfrrcncc or ecithout it, to draw a tangent to a giccn circlc. Closely connected with Prop. 16 are Props. 18 and 19, which state (Prop. 18), that the linc joining the ccntre of a circle to the point of contact of a tangent is pcipcntlicular to the tangent; and con- verse] y (Prop. 19), that the straight line through the point of contact of, and pcr1‘)cntlicttlar to, a tangent to a circle passes through the centre of the circle. §33. The rest of the book relates to angles co1111ccted with a circle, viz., angles which have the vertex either at the centre or on the cireiunfcrcnce, and which are called respectively angles at the centre and angles at the circumference. Between these two kinds of angles exists the important relation expressed as follows :— 1'rop. 20. The angle at the centre of a circle is double of the angle at the circlunfcrcnce on the same base, that is, on the same arc. This is of great importance for its consequences, of which the two following are the principal :—- Prop. :21. The anglcs in, the same segment of a circle arc cqual to one another; And Prop. 22. The opposite angles of any quadrilateral jigurc i-nscribcrl in a circle are together equal to two right angles. Further consequences are :— Prop. 23. On the same straight line, and on thc same side qf it, there cannot be two similar segments of circles, not coinciding ieilh ‘me another ; And Prop. 24. Similar segments qf circles on equal straight lincs are equal to one another. The problem Prop. 25, A scgmcnt of a circle being giccn to describe the circle of u‘/iirh it is a scgmcnt, may be solved much more easily by aid of the construction described in relation to Prop. 1, III., in §27. § 34. There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtcnding these arcs. They are expressed for angles, arcs, and chords in equal circles, but they hold also for angles, arcs, and chords in the same circle. The theorems are :— Prop. 26. In equal circlrs cqual angles stand on equal arcs, whether th-_'g be at the centres or circumferences ; Prop. 27 (converse to Prop. 26). In cqual circles the angles ichich stand on equal arcs arc equal to one another, uhether they bc at the centres or the circumferences; Prop. 28. In equal circles equal straight lincs (equal chords) cut of equal arcs, the greater equal to the greater, and the less cqual to the less ; Prop. 29 (converse to Prop. 28). In equal circlcs equal arcs are subtended by cqual straight lines. § 35. Other important consequences of Props. 20-22 are :— Prop. 31. In a circle the angle in a semicircle is a right angle ,- but the angle in a segment greater than a scmic-irclc is less than a right angle ; and the angle in a segment less than a semicircle is greater than a right angle ; Prop. 32. If a straight line touch a circle, and from the point, of contact a straight l inc be drawn cutting the circle, the angles irhich this line -makes ‘lt'lllt the line touching thc circle shall be equal to the angles "which are in thc al-tcrn_ate segments of the circle. ETRY 381 § 36. Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them :— Prop. 30. To biscct a given are, that is, to divide it into two equal parts ; Prop. 33. On a given straight line to describe a scgment of a circle containing an angle equal to a given rcctilineal anglc _; Prop. 34. From a giccn circle to cut of a segment containing an angle equal to a gircn rcctilincal anglc. § 37. If we draw chords through a point A within a circle, they will each be divided by A into two segments. Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken. The value of this rectangle changes, of course, with the position of A. A similar theorem holds if the point A be taken without the circle. On every straight line through A, which cuts the circle in two points B and L‘-, we have two segments AB and AL‘, and the rectangles contained by them are again equal to one another, and equal to the square on a. tangent drawn from A to the circle. The first of these theorems gives Prop. 35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36.' The first two theorems may be combined 11] one :— 'I‘II15o1:E.I.——[fthrough. a point A in the plane of a circle a straight line be drawn cutting the circle in B and 0', then the rectangle AB. .40 has a constant value so long as the point A be fired ; and if from A a tangent AD can be drawn to the circle, touching at D, then thc abore rectangle cquats the square on AD. Prop. 37 may be stated thus :— 'l‘1n50nE.I.—If from a point A n-ithout a circle a line be drawn cut- ting thc circle in B and C’, and another line to a point I) on the circle, and -if AB.AC=AD'-’, then the line AD touches the circle at D. It is not difficult to prove also the converse to the general pro- position as above stated. It may be expressed as follows :— If four points ABCI) be taken on the circumference of a circle, and if the lines A1)’, CI), produced if necessary, meet at E, then EA. EB = EC. ED; and conrcrsclg, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth. That a circle may always be drawn through three points, pro- vided that. they do not lie in a straight line, is proved only later on in Book IV. BOOK IV. § 38. The fourth book contains only problems, all relating to the construction of triangles and polygons inscribed in and circum- scribed about circlcs, and of circles inscribed in 01' circumscribed about triangles and polygons. They are nearly all given for their own sake, and 11ot for future use in the construction of figures, as are most of those in the former books. In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle. Instead, however, of saying that one figure is described about another, it is now generally said that the one figure is circumscribed about the other. 'e may then state the definitions 3 or 4 thus :— Dqfinition.——A polygon is said to be inscribed in a circle, and the circle is said to be circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the circle. And definitions 5 and 6 thus :— Definition.—..- polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the polygon are tangents to the circle. § 39. The first problem is merely constructive. It requires to draw in a given circle a chord equal to a given straight line, which is not greater than the diameter of the circle. The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the cireiunference. This may be said of almost all pro- blems in this book, especially of tlie next two. They are :— Prop. 2. In a given circle to inscribe a triangle cquiangular to a gicen triangle ; Prop. 3. About a giccn circle to circumscribc a triangle cqui- angular to a gii'-cn triangle. 3‘ 40. Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found. Prop. 4. To inscribe a circle in a gircn triangle. The result is that the problem has always a solution, viz., the centre of the circle is the point where the bisectors of two of the. nterior angles of the triangle meet. The solution shows, though Euclid does not state this, that tl1e problem has but one solu- tion ; and also, _
TIIEOR1:_I .——T he three biscctors qfthc interior an glcs of an 3/ triangle