Showing that a curve from vertex $A$ of $\triangle ABC$ to side $BC$ intersects a curve from $C$ to side $AB$ at least once

Asked May 24 '22 at 22:39

Active May 24 '22 at 23:35

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Suppose that we have a triangle $\triangle ABC$, and two continuous curves inside it: one starts at the vertex A and ends on the side $\overline{BC}$, and the other one starts at the vertex $C$ and ends on the side $\overline{AB}$. I have to show that both curves intersect at least one time.

This seems pretty straightforward to show using continuity, but I'm having a hard time formalizing this. Any help will be appreciated.

edited May 24 '22 at 23:35

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asked May 24 '22 at 22:39

Julian

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I think that you can use Jordan's theorem. – Roberto Vargas May 24 '22 at 23:40
@RobertoVargas, the OP doesn't say that the curves are simple. – lhf May 24 '22 at 23:48
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Related: https://math.stackexchange.com/questions/2429172/intersection-of-continuous-curves-in-a-square and https://mathoverflow.net/questions/35514/pair-of-curves-joining-opposite-corners-of-a-square-must-intersect-proof – lhf May 24 '22 at 23:49

Showing that a curve from vertex $A$ of $\triangle ABC$ to side $BC$ intersects a curve from $C$ to side $AB$ at least once

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