. The principles of projective geometry applied to the straight line and conic . tices F of There are associated with everypair of conies two real points atwhich the conies determine the sameinvolution. If the conies have four realcommon tangents, the points areany two of the points of inter-section of these common tangents. If the conies have no realcommon tangents, consider theenvelope of the common conjugatesof lines through a point A onone of the sides / of the common 270 Principles of Projective Geometry the common 8elf-conjugato triangleof the conies (1) and (2). Let the poles of the lin
. The principles of projective geometry applied to the straight line and conic . tices F of There are associated with everypair of conies two real points atwhich the conies determine the sameinvolution. If the conies have four realcommon tangents, the points areany two of the points of inter-section of these common tangents. If the conies have no realcommon tangents, consider theenvelope of the common conjugatesof lines through a point A onone of the sides / of the common 270 Principles of Projective Geometry the common 8elf-conjugato triangleof the conies (1) and (2). Let the poles of the line awith regard to (1) and (2) beA and A. These points are onEG, where FEG is the commonself-conjugate triangle. The polarsof P any point on a with respectto (1) and (2) will meet in somepoint P. self-conjugate triangle of the conies(i)and(2). Let the polars of the point Awith regard to (1) and (2) bea and a. These pass through F,where FEO is the common self-conjugate triangle. The poles ofp any line through A with respectto (1) and (2) will have as theirconnector some line Since the points A, A andalso the points E, G, F are onthe conic, which is the locus of P\this conic will break up into apair of lines, one of which isEAAG and the other FP. The polars of P with respectto (1) and (2) will pass through , if FP be a, the locusof P for different positions of Pon a is the line a. Hence thelines a and a form an involution Since the lines a, a and alsothe lines e, g, f are tangents tothe conic, which is the envelope ofp, this conic will break up into apair of points, one of which iseaag and the other fp. The poles oi p with respect to(1) and (2) will be on p. There-fore, if .//> be A, the envelope of jpfor different positions of ^ throughA is the point A. Hence thepoints A and A form an involution. Theorems concerning Two Conies 271 pencil. If FG is taken as a, thenEF will be a. Therefore FG andFE are conjugate elements of theinvohition. There will be a pairof dou
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
