An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer
page 29 of 156 (18%)
page 29 of 156 (18%)
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*27. Importance of the theorem.* The importance of this theorem lies in the fact that, _A_, _B_, and _C_ being given, an indefinite number of quadrangles _Kâ_, _Lâ_, _Mâ_, _Nâ_ my be found such that _KâLâ_ and _MâNâ_ meet in _A_, _KâNâ_ and _LâMâ_ in _C_, with _LâNâ_ passing through _B_. Indeed, the lines _AKâ_ and _AMâ_ may be drawn arbitrarily through _A_, and any line through _B_ may be used to determine _Lâ_ and _Nâ_. By joining these two points to _C_ the points _Kâ_ and _Mâ_ are determined. Then the line joining _Kâ_ and _Mâ_, found in this way, must pass through the point _D_ already determined by the quadrangle _K_, _L_, _M_, _N_. _The three points __A__, __B__, __C__, given in order, serve thus to determine a fourth point __D__._ *28.* In a complete quadrangle the line joining any two points is called the _opposite side_ to the line joining the other two points. The result of the preceding paragraph may then be stated as follows: Given three points, _A_, _B_, _C_, in a straight line, if a pair of opposite sides of a complete quadrangle pass through _A_, and another pair through _C_, and one of the remaining two sides goes through _B_, then the other of the remaining two sides will go through a fixed point which does not depend on the quadrangle employed. *29. Four harmonic points.* Four points, _A_, _B_, _C_, _D_, related as |
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