The most pressing issue with taking "lengths" between two points is taking into account the orientation of the displacement between them. The orthodoxy is to correct this problem by taking the absolute value of the displacement between two points as the "length" between them; this is not only clumsy but also highly lacking in foresight. If, at this point, you think I'm talking rubbish, I urge you to ponder the following: why, in statistics, do we take the square of residuals rather than their absolute values? Taking the absolute value of displacements inherently requires a framework of absolute value arithmetic, which at best requires some understanding of the dreaded "triangle inequality" and at worst producing the simplest of logical errors (e.g. if two points have a separation of \(2\) units and a third collinear point is separated from one of them by \(1\) unit, calculating the separation between this point and the other point gives two different answers; RED FLAG ALERT!).
Consider the following alternate explanation: rather than taking the absolute value of displacements, why not square them? Sounds easy enough, doesn't it? In fact, it removes a lot of the issues that we faced earlier; first, and most importantly, of all, the orientation problem is gone (HOORAY!). If you have a basic understanding of linear algebra, you then would understand that squaring displacements is equivalent to taking the dot product of the displacement with itself. In foresight, a geometer would realise instantly that Pythagoras' theorem is heavily simplified to just a simple arithmetical result. So, you may ask now, what price do we pay for living in this "parallel universe"?
Let me just shatter your so-called mathematical bravado first by saying that in this domain which you insist on calling it an "alternate reality" (what utter BS, let's be frank), all our logical flaws are gone. Now, this "price" we pay; it's not really a big price to pay, unless you're infuriatingly lazy with computations and just looking for quick outs to proofs by clever tricks. Let me put it this way: a big benefit of working with squares of displacements is that all the geometrical results we can obtain are algebraic in nature, i.e. in polynomial form, but the (not-so-)big drawback is that the degree of these polynomial results can get exponentially large (that's if you're not careful).
In fact, let us illustrate this with one key result. We know that we can "add" "lengths" by a simple arithmetical relation; given three points \(A,B,C\) with "lengths" \(l_1 \equiv l(A,B)\), \(l_2 \equiv l(A,C)\) and \(l_3 \equiv l(B,C)\), then without knowledge of the orientation of the three points, we can deduce that one of the following three results are true:
\( l_1 + l_3 = l_2 \\ l_1 + l_2 = l_3 \\ l_2 + l_3 = l_1 \)
What happened if we squared these "lengths"? Suppose \(Q_i \equiv l_i^2\) for \(i=1,2,3\); then squaring both sides of one of these equations gives
\( Q_1 + Q_2 - Q_3 = 2l_1l_2 \),
which, when squared again, gives
\( (Q_1 + Q_2 - Q_3)^2 = 4Q_1Q_2 \).
We can expand the left-hand side of this equation and rearrange the whole equation to obtain a more symmetric form of this relation:
\( (Q_1 + Q_2 + Q_3)^2 = 2(Q_1^2 + Q_2^2 + Q_3^2) \).
This relation is known in RT circles as the Triple quad formula, and is the relation that links the pairwise (squared) separation between three points on a line, which we will now call the quadrance (which can be thought of classically as the squared "length"). The symmetric form of this relation implies and confirms that the framework of squaring "lengths" is orientation-free and (in foresight) opens the door to creating more general frameworks of geometry just by altering the definition of our dot product (understanding of linear algebra required; will explain in another time).
Wait, so the relationship between the pairwise separation of three points is a QUADRATIC one? But we were able to just add two "lengths" together; it's the easy, and intuitive thing to do... Well, yeah, but I just explained all the logical issues. Sure, it's a price to pay, but just look at the formula: IT'S SYMMETRIC! I know you won't get that by taking absolute values of displacements, that's for sure. So ultimately, it's not a big price to pay, since we actually GAIN information by squaring the "lengths", instead of losing information; I refer to Wildberger's (text)book on Rational Trigonometry to illustrate more of this beauty.
One more thing I should mention is whether there is a derivation of this formula without any reference to "lengths". Of course there has to be, right? It's not like Norman just plucked these formulas on a whim... I will say, for the record, that to prove this relation we can just cook up three collinear points, compute their quadrances and substitute them into the equation and let magic work on itself. But, of course, there has to be some sort of geometrical intuition to this result. For this, I refer you to his video on this stuff, linked here. And, at this point (if you have the galls to get here), I shall leave you to ponder your thoughts on this. Until next time...
(Stay tuned for the next installment, where I talk about the Quadruple quad formula & higher-dimensional variations of it!)