Generalized butterfly theorem

Given circle \((O)\) with chord \(AB\), let \(I\) be a point on any position of the chord \(AB\) (except \(A\) and \(B\)). Draw two more chords, \(CD\) and \(EF\) so that the chord \(CE\) does not intersect the chore \(AB\) of the circle. \(CF\) and \(DE\) intersect \(AB\) at \(M\) and \(N\) respectively. Prove that: \(\frac{AM\times IB}{IM}=\frac{BN\times IA}{IN}\) and \(\frac{1}{IA}+\frac{1}{IN}=\frac{1}{IB}+\frac{1}{IM}\)

Obviously, with the case \(IM=IN\), we can solve it by using the butterfly theorem.

I think the first problem might be related to the second problem as both equalities are quite similar to each other, I have attempted to use the Haruki lemma to prove that \(AM\times IB=IM\times AG\) and \(BN\times IA=IN\times BH\) but I'm stuck at this point, so proving \(AG=BH\) would be neccessary (I draw two extra circles to prove the Haruki lemma as above).

Note by Trần Thúc Minh Trí
9 months, 2 weeks ago

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