6. PASCAL’S THEOREM

Cross Ratio on a Line 
(A, B; C, D) = (1, 3; 2, 4) = 
Cross Ratio of a Pencil Let a, b, c, d be four lines passing through point O, and draw an arbitrary line (transversal) not through O meeting them in points A, B, C, D as in the diagram. We prove:
Proof Draw through B a line parallel to a meeting c and d in points P, Q respectively, as in the diagram. We assume that lines a and PQ are directed in the same sense. Triangles ACO, BCP are similar, so . Similarly, triangles ADO and BDQ are similar, giving . Hence . It follows that the cross ratios of points on all transversals through the given point B are equal. In fact, this value doesn't depend on the actual position of B since the value on the line parallel to a remains constant for all positions of B: both numerator and denominator of the fraction are musltiplied by the same constant which then cancels out. 
A possible converse? Using the adjacent diagram we have hown that the value of the cross ratio (A, B; C, D) is independent of the position of the transversal. We can prove an important partial converse result. Use this diagram.
Proof Let BB' and CC' meet in point O, and let OD meet line AB'C' in D". 
The easiest way to see this is probably to use coordinates. With the obvious notaion we have whence we obtain d' = d". 
Proof of Pascal’s Theorem We first observe that in the adjacent diagram, the green pencil and the yellow pencil are actually congruent – the angles between corresponding lines are equal, using the subtending property of chords in a circle. Obviously then theses two pencils have the same cross ratio. The value of this common cross ratio is the same as the cross ratio of the points on any transversals. So we get 
Extensions 1. The cross ratio of four points on a line is worth investigating in its own right. For example, if the cross ratio is negative, what can you say about the two pairs of points? You should be able to show that the value of the cross ratio of four points remains unchanged when two pairs of points are interchanged. For looking up ... 
Hints and Solutions
1. The cross ratio of four points on a line is worth investigating in its own right. For example, if the cross ratio is negative, what can you say about the two pairs of points? You should be able to show that the value of the cross ratio of four points remains unchanged when two pairs of points are interchanged. 