Total least squares is pretty bog standard in statistics and has a lot of literature, including monographs and text books. You are correct about the Pythagoras

Total least squares is pretty bog standard in statistics and has a lot of literature, including monographs and text books. You are correct about the Pythagoras theorem and by virtue of that TLS has close connections with PCA, in fact, once you have the PCA model you can derive the TLS coefficients from the PCA parameters. The tricky bit is that in TLS the number of nuisance parameters grows with the data so it wouldn't be immediately obvious that estimates would converge. It turns out that it does.