maximum Corelation coefficient - how the numarator and denominator becomes equal?

Pearsons corelation coefficient is defined as follows

$$ r_{x,y} =frac {sum (x_i -bar X)(y_i – bar Y)}{sqrt{sum(x_i-bar X)^2} sqrt{sum(y_i-bar Y)^2}} $$.

Now, maximum magnitude of a correlation coefficient is known to be $1$;
which is possible only if

$$Numarator = denominator$$,

$$or, {sum (x_i -bar X)(y_i – bar Y)}={sqrt{sum(x_i-bar X)^2} sqrt{sum(y_i-bar Y)^2}} … (1) $$;

Now; if the dataset is perfectly corelated, then how or why the relation $… (1)$ works? or in other words, the denominator or numarator becomes equal?

Samely the alternate equation,

$$r_{xy}= frac {nsum X_iY_i- (sum x_i)(sum y_i)} {sqrt{nsum x_i^2 -(sum x_i)^2} sqrt{nsum y_i^2-(sum y_i)^2}}$$

now, the coefficient of correlation will be =1 only if


$$or,{nsum X_iY_i- (sum x_i)(sum y_i)} = {sqrt{nsum x_i^2 -(sum x_i)^2} sqrt{nsum y_i^2-(sum y_i)^2}} … (2) $$

If the dataset is perfectly correlated ; then how to tell relation $… (2)$ will work?

So in brief my question is ….

I am looking for necessarily intuitive explanation of what these mathematical terms mean; such as it is quite clear that ${sum (x_i -bar X)(y_i – bar Y)}$ is a sum of areas made by the data points which comes from covariance. But why this exact same area will be cancelled out by the denominator, that I could not understand by any way.

What I tried already:

  1. Searched Google and Stackexchange by search terms.
  2. Tried youtube videos like this, this etc.
  3. Have visited several internet articles but the articles are appearently lacking visual or intuitive explanations.

Cross Validated Asked on November 21, 2021

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