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.
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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Three angles property ![[0_0]](ThreeAngles_files/ThreeAngles_0_0_0.png)
![[0_1]](ThreeAngles_files/ThreeAngles_0_0_1.png)
![[1_0]](ThreeAngles_files/ThreeAngles_0_1_0.png)
![[1_1]](ThreeAngles_files/ThreeAngles_0_1_1.png)