. The principles of projective geometry applied to the straight line and conic . ith a line, drawn in a fixed direction through the opposite vertex, is aconic. It should be noticed that the sides of triangles given in species determine threeprojective ranges on the line at infinity. (1) Let the vertex A be fixed and let Bmove along the straight line I. Then CBdetermines on the line at infinity a range pro-jective with that determined by AB, and there-fore with the range described by B on I. Hencethe envelope of CB is a parabola touching theline I. Similarly the parabola touches the line inon w
. The principles of projective geometry applied to the straight line and conic . ith a line, drawn in a fixed direction through the opposite vertex, is aconic. It should be noticed that the sides of triangles given in species determine threeprojective ranges on the line at infinity. (1) Let the vertex A be fixed and let Bmove along the straight line I. Then CBdetermines on the line at infinity a range pro-jective with that determined by AB, and there-fore with the range described by B on I. Hencethe envelope of CB is a parabola touching theline I. Similarly the parabola touches the line inon which (Addendum 6 (a)) C moves. If mand I meet at T the points C, T, B, A areconcyclic and therefore A is the focus of theparabola. Example (3), Chapter xiii. (2) If a line through B drawn in a fixeddirection meets AC at P, P is the point ofintersection of rays of two projective pencils,whose vertices are at infinity in the givendirection and at A. Hence the locus is aconic through A. *In Addendum G some important theorems connected with triangles, given in species,are Loci and Envelopes 249 6. Find the envelope of a side of a triangle given in species and inscribed in agiven triangle. If the triangle AEC be inscribed in the triangle ABC^ the circumcircles ofA CB\ BCA and CAB meet at a fixed point 0, and the triangle OAB is given inspecies (Addendum 6 {b)). Since 0 is fixed and A and B lie on two fixed linesthe envelope of AB is a parabola of which 0 is the focus. 7. If a series of similar figures are described, three corresponding points ofwhich are situated on the sides of a triangle, then the loci of all other points ofthe figures are straight lines, and the envelopes of all hues are parabolas which havethe same point for focus. In the last example let P be a variable point such that ABCP is alwayssimilar to a given figure. Then the triangle OAF is given in species, 0 is fixedand A moves along a fixed line. Hence the locus of P is a fixed line. If i*, </ be any tw
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
