Let $\gamma$ be a Jordan curve and suppose two arbitrary points $A$ and $B$ are given. Then does there exist a point $C$ on $\gamma$ such that $ABC$ form a Isosceles triangle with sides $AC=BC$?
2026-03-28 01:06:38.1774659998
Does there exist a point $C$ for given two point $A$ an $B$ on a Jordan curve such that $ABC$ form a Isosceles triangle?
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Yes. To find the point $C$ consider the bisector $s$ of the segment $AB$. The curve $\gamma$ joins $A$ and $B$. Since $\gamma$ is a Jordan curve, it must intersect $s$ at two or more points (because $A$ and $B$ are in different semiplanes with respect to $s$). Choose one of these points that does not lie on the segment $AB$ and you are done.