[alogo] Three angles property

Three angles A,B,C have a common chord OO'. Then they have pairwise three other common chords MM',NN',PP' intersecting at a point Q.

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Below the equation between symbols means that the corresponding line-bundles define the same cross-ratio. Further MA*PN denotes the intersection point of the two lines.
O'(A,M',N',O) = O'(N,P',C,O)  => M(A,M',N',O) = P(N,P',C,O) but {P,O,M} on a line (see CrossRatioLines.html ) => the intersections of the other rays are on a line:
MA*PN=N, MM'*PP'=Q, MN'*PC=N', hence NN' passes through Q.
Remark-1 Q is the pole of line OO' of the conic passing through {A,B,C,O,O'}.
Remark-2 When B moves parallel to AC then Q remains fixed.

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