Does the equality coefficients of linear regression of X onto Y and Y onto X imply coincidence of the lines?

Question

Let's assume that we consider the model without an intercept $\hat{y_i} = x_i\hat{\beta}$. So, the formula for $\hat{\beta}$ is $\frac{ \sum\limits_{i=1}^n x_i y_i}{\sum\limits_{j=1}^n x_j^2}$. I know that $\hat{\beta}_{XY}$ = $\hat{\beta}_{XY}$ if and only if $ \sum\limits_{j=1}^n x_j^2 = \sum\limits_{j=1}^n y_j^2$ (Here, $\beta_{XY}$ means regression coefficient of $X$ onto $Y$). If some of the squares indeed equal, we have that $\hat{\beta}_{XY}$ = $\hat{\beta}_{XY}$, however I don't understand whether it implies the coincidence of the lines.

Answer 1

No. Assume that slope is $\sqrt{3}$. As you can see, the lines $y = \sqrt{3}x$ and $x = \sqrt{3}y$ do not coincide.