In this work, we give answers to questions regarding the existence and the number of equilateral triangles on Jordan curves. The main part of the work gives these results in the plane. We define the intersection number of two functions and take a look at triods. Using these concepts we show that all but two points on a Jordan curve are vertices of some equilateral triangle on this curve. In the second part we generalize the results from the plane to spaces of higher dimensions. At the end we take a look at some of the results on the topic of squares on Jordan curves. We also show that there always exists a rectangle on a Jordan curve.
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