It was established by *TMA*, @WillSawin, and @DouglasZare, in their responses to
the MO question,
"[Thales' semicircle theorem in higher dimensions](https://mathoverflow.net/q/201281/6094),"
that the natural generalization of Thales' semicircle theorem to $\mathbb{R}^3$
is false.

> ***Q***. Is there *any* natural measure/quantity that is preserved as a point $p$
varies over a unit hemisphere centered at $o$,
and somehow projects a geometric figure on the base unit circle of the hemisphere?

It is a bit surprising to me that the straightforward generalizations fail.
One obvious and useless answer to ***Q*** is that $||p-o||=1$.
Perhaps ***Q*** is unanswerable, but I have the sense that something more substantive
than the radius should be preserved...