Give an example of a continuous curve $\gamma$ in the Hilbert space (i.e., an image of a continuous map $\phi : [0,1] \rightarrow H)$ with the following “miraculous” property: for every three points $A,B, C\in \gamma$, the triangle ABC is right-angled.
This is a homework problem. I am not asking for a solution. But I am just requesting if you can point me to a direction in which I should proceed. Because I am quite clueless how such a curve is possible in an Euclidean space.