1) Consider the quadrilateral AG''BG'. The intersection point I of the diagonals G'G'', FH is harmonic conjugate to C''', with respect to F,H, C''' being the intersection point of AB, FH. But since (C*A, C*B*, C*C, C*H) is a harmonic bundle, the same is true for the intersection point of FH with CC*. Hence FH, G'I and CC* intersect at a point I.

2) It follows that (AC*, AI, AC', AH) is a harmonic bundle, hence (C*IA*C')=-1. This implies that (GA*, GC', GI, GC*) is a harmonic bundle of lines hence line GI is the polar of C*, proving the collinearity.

Note that in the case D, E coincide i.e. (e) is tangent to (c), then I is the contact point of (c) and (e) and this is collinear with F and H. This property is used in the construction of the family of conics through four points and tangent to a given line, discussed in the file AllConicsCircumscribed.html .

Harmonic.html

ThreeCollinearPts.html

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