. The principles of projective geometry applied to the straight line and conic . BB be 0 and R Join OR. Then the points P, F, ifreal, in which OR meets theconic are the common pair ofconjugates of the two involutions.(Art. 95 (c).) By the correlative method the two tangents, which are a pair of conjugates intwo invohitions of tangents to the conic, may be constructed. 112. Method of false positions. In attempting to solve ageometrical problem it frecjuently happens that the solution of theproblem is found to depend on finding the configuration for which twopoints coincide. Such points
. The principles of projective geometry applied to the straight line and conic . BB be 0 and R Join OR. Then the points P, F, ifreal, in which OR meets theconic are the common pair ofconjugates of the two involutions.(Art. 95 (c).) By the correlative method the two tangents, which are a pair of conjugates intwo invohitions of tangents to the conic, may be constructed. 112. Method of false positions. In attempting to solve ageometrical problem it frecjuently happens that the solution of theproblem is found to depend on finding the configuration for which twopoints coincide. Such points are frequently situated on some givenline or conic and in this case it often happens that for differentpositions of one of the points it may be proved that the second pointdescribes a range projective with that described by the first. In suchcases the problem can immediately be solved. Any three positionsof the first point may be taken and the three corresponding positionsof the second. The projective ranges are then completely determined 238 Principles of Projective Geometry. ^u ranges, and the self- and their self-corresponding points can (by Arts. 109 and 110) be atonce constructed. These self-corresponding points give the requiredconfiguration. The following are instances of the application of thismethod : (rt) To itiscrihe a triangle in a tjiven triangh- so that its sides may pass throughthree given points. Let ABC be tlie given triangle A and l\ r, W the given U draw a transversal tomeet BC and BA in P and Q. Joinr to (^ to meet AC in R. Join Wto J{ to meet BC in *S. Then the range described by Pis projective with the range describedby Q. This is projective with therange described by E, which isprojective with the range described by *S. Therefore P and *S describe two superposed projecticorresponding points of these give the solution of the problem. (h) To inscribe a square in a given triangle in such a way that one of its sidesmay lie along a .side of the
Size: 2401px × 1041px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
