543. If A, C, B, D form a harmonic range, and O be the mid point of AB, then OA OB OC. OD. = For AD/DB= AC/BC. :. (AD + DB)/{(AD – DB) S :: OB2 = OC.OD. 544. Theorem. If four concurrent straights cut any transversal in a harmonic range, they will cut every transversal in a harmonic range. FIG. 236. Proof. Through B and B' draw BT and B'T' to AA'S, and meeting SC in T, T and SD in T2, T. Then since. BT, BT,,.. B'T' = B'T. = .. A'C'B'D' is a harmonic range. 545. If A, C, B, D is a harmonic range, SA, SC, SB, SD is a harmonic pencil, and SC, SD are harmonic conjugates of SA, SB. 546. We have shown that the arms of any angle form with its internal bisector and its external bisector a harmonic pencil. 547. If, in a harmonic pencil, one element bisect the anglebetween two conjugates, then it is perpendicular to its conjugate. 548. If in a harmonic pencil one pair of conjugates be at right angles, then these are the internal and external bisectors of the angle between the other pair. 549. Theorem. If two harmonic ranges are taken, one in each of two straights, and if three of the four projectionstraights are concurrent, then so are the four. Proof. For the three concurrent projection-straights and the straight from their bearer through one of the fourth points form a harmonic pencil; so this latter straight contains also the other fourth point. 550. Corollary. If two corresponding points coincide in the cross of the two straights, then one projection-straight being free, the other three are always concurrent. CHAPTER III. PRINCIPLE OF DUALITY. 553. In a pencil consisting of straights through one fixed point, any one of the straights may be called an element of the pencil, or a straight on the fixed point or bearer. In this sense, we say not only that points may lie on a straight, their bearer, but also that straights may lie on a point, their bearer, meaning that the straights pass through this point. 554. In most cases we can, when one figure is given, construct another, such that straights take the place of points in the first, and points the place of straights. Thus from a definition or a theorem we can obtain another by interchanging point and straight, cross and join, range and pencil, or by similar interchanges. 555. A figure regarded as consisting of a system of straights. crossing in points will thus give a figure which may be regarded as a system of points joined by straights; and in general with any figure coexists another having the same genesis from these elements, point and straight, but that these elements are interchanged. Any descriptive theorem or theorem of position concerning. one, thus gives rise to a corresponding theorem concerning the other figure. 556. Figures or theorems related in this manner are called dual figures or dual theorems. 557. This correlation of point and straight is termed a principle of duality. 558. Each of two descriptive theorems so correlated is said to be the dual of the other; and it will be found that if any descriptive property is demonstrated, its dual also holds. 559. Since capitals mean points, and two fix a straight, their join; so small letters may denote straights, and two will fix a point, their cross. Thus AB denotes the straight which is the join of the points A and B; while ab denotes the point which is the cross of the straights a and b. 560. In plane geometry to all points on a straight the dual figure is all straights on a point. 561. A sect, AB, may be considered a range containing the initial point A of the sect, its final point B, and all intermediate successive positions of the generating point. 562. The figure dual to sect AB is ab, that piece of a pencil containing the initial straight a of the angle, its final straight b, and all intermediate positions of the generating straight. DUAL THEOREMS. 563,. If two harmonic ranges are taken, one in each of two straights, and if three of the four joins of corresponding points are concurrent, then so are the four. 563'. If two harmonic pencils are such that three of the four crosses of pairs of corresponding straights are straight, then so are the four. CO PRINCIPLE OF DUALITY. 125. 564'. If two straights, one in each of two harmonic pencils, are coincident, then the three crosses of the other 564,. If two harmonic ranges are taken one in each of two straights, and two corresponding points coincide in the cross of the straights, three pairs of straights are. then the other three projec- costraight. tion-straights are concurrent. 5642 In Symmetry. 564". In Symcentry. 1. The symcenter is with regard to itself. 2. Every st' on the symcenter is to itself; and inversely. 3. The join of two p'ts is on the symcenter. 4. Two st's and any st' on the symcenter bound equal 48. 5. Two p'ts make = sects with the symcenter. 6. Two st's have equal 1 from the symcenter. 7. The join of two p'ts is to the join of the two p'ts. 8. Two and a transversal. 9. Two and two st's on the symcenter. 10. Two pairs of ||3. CHAPTER IV. COMPLETE QUADRILATERAL AND QUADRANGLE. 565, A system of four straights, no three concurrent, and their six crosses is called a complete quadrilateral, or tetragram. 566,. The four straights are called the "sides" of the quadrilateral; and the six crosses, the vertices. 567,. Two vertices which do not lie on the same "side" are called opposite vertices. There are three pairs. 568,. The three straights joining opposite vertices are called diagonal straights, and the triangle formed by the diagonal straights is called the diagonal triangle of the complete quadrilateral. 565'. A system of four points, no three costraight, and their six joins is called a quadrangle, or tetrastim. 566'. The four points are called the summits of the quadrangle, and their six joins the connectors. 567'. Two connectors which do not pass through the same summit are called opposite connectors. There are three pairs. 568'. The three crosses of opposite connectors are called diagonal points, and the triangle determined by the diagonal points is called the diagonal triangle of the quadrangle. 126 |