关于PuMAC Combinatorics A 2022 Problem 2官方解答高亮部分的理解问询

这是PuMAC Combinatorics A 2022的Problem 2，题目内容如下：

Ten evenly spaced vertical lines in the plane are labeled ℓ₁, ℓ₂, . . . , ℓ₁₀ from left to right. A set {a, b, c, d} of four distinct integers a, b, c, d ∈ {1, 2, . . . , 10} is squarish if some square has one vertex on each of the lines ℓₐ, ℓᵦ, ℓc, and ℓd. Find the number of squarish sets.

我试着用坐标几何的思路来解这道题，但需要处理的方程实在太多，最后没能顺利解出来。

后来我找到了官方解答，它用了一个非常巧妙的方法，一下子把问题简化了，但我实在搞不懂里面的高亮部分。官方解答的相关内容如下：

Without loss of generality, assume that a < b < c < d. Then, it is easy to see that {a, b, c, d} is squarish if and only if the distance...

备注：内容来源于stack exchange，提问作者Aashita