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4}It is possible, by a sequence of not more than three perspectivities, to relate any three distinct collinear points to any other three distinct collinear points.
Proof:
If the two triads, ABC and A'B'C' are on distinct lines, let R,S,C0 denote the points where the respective lines AA', BB', BA' meet CC'. Then ABC and A'B'C' are related by the sequence of two perspectivities ABC =/\R= A'BC0 =/\S= A'B'C'.
Xah's note: this will work for any collinear ABC and A'B'C', regardless of order. Look at the way R is defined, ABC =/\R= A'BC0 will always be true. Similarly for S of A'BC0 =/\S= A'B'C'. Therefore the order of ABC or A'B'C' doesn't matter. And I think we can also reach the conclusion by looking at the cyclic order property on point on a line.
If A and A' coincide, we merely use the perspectivity from S. If the two triads are on one line, we use a quite arbitrary perspectivity ABC =/\= A1B1C1 to obtain a triad on another line and then relate A1B1C1 to A'B'C' by the abvoe construction.n.n.n. WIND STR#l
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