geometry – Explanation of Cartesian formula for circumcenter

On Wikipedia there is a Cartesian formula for the circumcenter of a triangle. That is, given points $$A$$, $$B$$ and $$C$$ in $$mathbb{R}^2$$, find point $$U$$ such that $$d(A,U)=d(B,U)=d(C,U)$$. The formula, as stated on Wikipedia, is very algebraic, but I found that it can be rewritten as follows.

Define $$X:=(A_x,B_x,C_x)$$, $$Y:=(A_y,B_y,C_y)$$, $$V:=(1,1,1)$$ and $$L:=(|A|^2,|B|^2,|C|^2)$$. Then define the matrices $$M_D:=(U,X,Y)$$, $$M_X:=(U,X,L)$$ and $$M_Y:=(U,L,Y)$$. Then we have $$U=(|M_Y|,|M_X|) / (2|M_D|).$$

As I am writing a program, I very much enjoy this elegant form. However, such a nice formula needs a nice explanation if you ask me. Can anyone come up with one? I am hoping for an explanation that does not rely on any of the algebraic properties of the determinant and only uses its geometric interpretation based on measures.