# Why the geometrical mean of the slope of the two regression lines can be a measures linear correlation between two variables?

The Pearson correlation coefficient is the geometrical mean of the slopes of the two regression lines. For me it is difficult to understand how a geometrical mean of the slope of the two regression lines can be a measure of the linear correlation between two variables X and Y.

I found this paper fantastic, Bravais-Pearson and Spearman correlation coefficients: meaning, test of hypothesis and confidence interval by Artusi and coll., in explaining the Pearson coefficient, but I have difficulties in doing the last deduction.

Let's regress $$Y$$ on $$X$$ and write $$b_{0x} + b_{1x} x$$ for the regression line, where $$b_{1x}$$ is the slope. Then Pearson correlation coefficient is $$r = b_{1x} \frac{s_x}{s_y}$$, $$s_x$$ and $$s_y$$ being the standard deviations of $$X$$ and $$Y$$, respectively. If, on the contrary, we regress $$X$$ on $$Y$$, the regression line is defined by $$b_{0y} + b_{1y} y$$, thus $$r = b_{1y} \frac{s_y}{s_x}$$.
Then it follows that $$r^2 = b_{1x} b_{1y}$$, i.e. $$r = \sqrt{b_{1x} b_{1y}}$$, which is the geometric mean of the two slopes.