Origin of “on thin ice”
Solution 1:
According to the OED, for the specific phrase "on thin ice", they state:
"colloquial (originally U.S.). on thin ice: in a precarious or risky situation."
The earliest example they give for "on thin ice" is 1871, from "Dartmouth" (I don't know what that is):
"But dodging this point by the word possibility, he finds himself on thin ice and skates fast to the conclusion."