# In a mn and mn in a kn and kn in b lm and lm in b ln

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in A, MN and M'N' in A', KN and K'N' in B, LM and L'M' in B', LN and L'N' in C,and KM and K'M' in C', then, if A, A', B, B', and C are in a straight line, the point C' also lies onthat straight line.The theorem follows from Desargues's theorem (Fig. 32). It is seenthatKK',LL',MM',NN'all [pg 72]meet in a point, and thus, from the same theorem, applied tothe trianglesKLMandK'L'M', the pointC'is on the same line withAandB'. As in the simplercase, it is seen that there is an indefinite number of quadrangles which may be drawn, two sidesof which go throughAandA', two throughBandB', and one throughC. The sixth side must thengo throughC'. Therefore,122.Two pairs of points, A, A' and B, B', being given, then the point C' corresponding to anygiven point C is uniquely determined.
The construction of this sixth point is easily accomplished. Draw throughAandA'any two lines,and cut across them by any line throughCin the pointsLandN. JoinNtoBandLtoB', thusdetermining the pointsKandMon the two lines throughAandA', The lineKMdetermines thedesired pointC'. Manifestly, starting fromC', we come in this way always to the same pointC.The particular quadrangle employed is of no consequence. Moreover, since one pair of oppositesides in a complete quadrangle is not distinguishable in any way from any other, the same set ofsix points will be obtained by starting from the pairsAA'andCC', or from the pairsBB'andCC'.123. Definition of involution of points on a line.Three pairs of points on a line are said to be in involution if through each pair may be drawn apair of opposite sides of a complete quadrangle. If two pairs are fixed and one of the third pairdescribes the line, then the other also describes the line, and the points of the line are said to bepaired in the involution determined by the two fixed pairs.[pg 73]FIG. 33124. Double-points in an involution.The pointsCandC'describe projective point-rows, asmay be seen by fixing the pointsLandM. The self-corresponding points, of which there are twoor none, are called thedouble-pointsin the involution. It is not difficult to see that the double-
points in the involution are harmonic conjugates with respect to corresponding points in theinvolution. For, fixing as before the pointsLandM, let the intersection of thelinesCLandC'MbeP(Fig. 33). The locus ofPis a conic which goes through the double-points,becausethepoint-rowsCandC'areprojective,andthereforesoarethepencilsLCandMC'which generate the locus ofP. Also, whenCandC'fall together, thepointPcoincides with them. Further, the tangents atLandMto this conic described byPare thelinesLBandMB. For in the pencil atLthe rayLMcommon to the two pencils which generatethe conic is the rayLB'and corresponds to the rayMBofM, which is therefore the tangent lineto the conic atM. Similarly for the tangentLBatL.LMis therefore the polar ofBwith respect tothis conic, andBandB'are therefore harmonic conjugates with respect to the double-points. The

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Term
Fall
Professor
Dr.Movahed
Tags
Projective geometry, Desargues, Synthetic Projective Geometry