# 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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